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Full text of "The elements of algebra"

llA'^ 



LIBRARY 

OF THK 

University of California. 



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http://www.archive.org/details/e1ementsofalgebr00lillrich 



THE 



Elements of Algebra 



BY 



GEORGE LILLEY, Ph.D., LL.D. 



EX-PRESIDENT SOUTH DAKOTA AGRICULTURAL COLLBOB 



OF THE ^ 

UNIVERSITY 




SILVER, BURDETT & COMPANY 
New York . . . BOSTON . . . Chicago 
1894 






Copyright, 1892, 
By Silver, Burdett and Company. 



mniijcrsitg lirrss : 
John Wilson and Son, Cambridqk, U.S.A. 



PREFACE. 



Algebra is a means to be used in other mathematical 
work; it develops the mathematical language, and is the 
great mathematical instrument. If the student would be- 
come a mathematician, he must understand this language 
and possess facility in handling the various forms of literal 
expressions. 

Attention is called to the sequence of subjects as herein 
presented. Involution is introduced as an application of 
multiplication, evolution as an application of division, and 
logarithms as an application of exponents. Throughout the 
book the student is led to see that one subject follows as an 
application of another subject. The beginner is led to see at 
the outset that Algebra, like Arithmetic, treats of numbers. 

Algebraic terms and definitions are not introduced until 
the student is required to put them into actual use. Correct 
processes are cleariy set forth by carefully prepared solutions, 
the study of which leads the pupil to discover that method 
and theory follow directly from practice, and that methods 
are merely clear, definite, linguistic descriptions of correct 
processes. 

The book is sufficiently advanced for the best High Schools 
and Academies, and covers sufficient ground for admission to 
any American College. 

Great care has been given to the selection and arrangement 
of numerous examples and problems. These have been, for 

18:5963 



iv PREFACE. 

the most part, tested in the recitation-room, and are not so 
difficult as to discourage the beginner. 

It remains for the author to express his sincere thanks 
to W. H. Hatch, Superintendent of Schools, Moline, 111. ; to 
Professor W. C. Bojden, Sub-Master of the Boston Normal 
School, Boston, Mass. ; and to O. S. Cook, connected with 
the literary department of Messrs. Silver, Burdett & Co., for 
reading the manuscript and for valuable suggestions. 

GEORGE LILLEY. 

Pullman, Washington, June, 1892. 



PREFACE TO THE SECOND EDITION. 

In this edition the typographical errors have been cor- 
rected, and a page of examples has been added to Chapter 
XXVII ; also, the exercises have been carefully revised and 
corrected. Answers to the examples and problems have been 
prepared, and are bound in the book, or separately in flexi- 
ble cloth covers. The answer-book is furnished for the use 
of the class only on application of teachers to the publishers 
for it. The publishers and the author desire to express their 
appreciation of the very favorable reception accorded to the 
first edition. 

September, 1894. 



CONTENTS. 



CHArrEK PACK 

I. First Principles l 

II. Algebraic Addition 19 

III. Algebraic Subtraction 27 

IV. Algebraic Multiplication 35 

V. Involution 52 

VI. Algebraic Division 60 

VII. Evolution 79 

VIII. Use of Algebraic Symbols 99 

IX. Simple Equations 104 

X. Problems Leading to Simple Equations 109 

XI. Factoring 119 

XII. Highest Common Factor 141 

XIII. Lowest Common Multiple 155 

XIV. Algebraic Fractions '164 

XV. Fractional Equations •. . . 201 

XVI. Simultaneous Simple Equations 215 

XVII. Problems Leading to Simultaneous Equations . . 238 

XVIIL Exponents 248 

XIX. Radical Expressions 263 

XX. Logarithms 296 

XXI. Quadratic Equations 312 



vi CONTENTS. 

CHAPTEK PAGE 

XXII. Equations which may be Solved as Quadratics . 330 

Theory of Quadratic Equations 339 

XXIII. Simultaneous Quadratic Equations 345 

XXIV. Indeterminate Equations 355 

XXV, Inequalities 363 

XXVI. Series 373 

Arithmetical. 373 

Geometrical 379 

Harmonical 384 

XXVII. Ratio and Proportion 388 



APPENDIX 401 



INDEX TO DEFINITIONS. 



PAOI 

Algebra 118 

Binomial 46 

Coefficient 20 

Equation, Biquadratic . 334 

" Degree of, Roots of 35, 108 

" Exponential 309 

Literal 206 

" Syminetriciil 347 

Expression, Algebraic 90 

" Compound 23 

** Homogeneous 349 

** Imaginary 286 

** Irrational 263 

** Mixed 164 

" Simple 21 

Factor 119 

Figures, Subscript 227 

Fraction, Complex 188 

Continued 190 

Identities 104 

Index 79 

Mean, Arithmetical 378 

" Geometrical 383 

" Harmonical 385 

Nlonoraial 21 

Multiple ,155 

" Common 155 

Multiplication, Algebraic 38 



Vlll INDEX TO DEFINITIONS. 

PAGE 

Numbers, Algebraic, Absolute ... 14 

" Known 107 

" Negative 11 

" Scale of 12 

" Unknown 107 

Polynomial 23 

Power 35 

Progression, Arithmetical 373 

" Geometrical 379 

" Harmonical 384 

Quantity <.....• 389 

Keciprocal ^8 

Roots 79, 340 

Signs, Algebraic 13 

" Double 80 

" Law of . 38, 61 

" Radical 79 

Subtraction ' 34 

Surd, Similar 263 

" Entire, Mixed 264 

" Quadratic 290 

Symbols of Abbreviation 7 

" of Aggregation, of Relation 6 

" of Operation 1, 99 

Terms 3, 90 

" Like 20 

Term, Absolute 207 

" Degree of. Dimension of 345 

Value, Absolute • < 14 

** Numerical 9 




ELEMENTS OF ALGEBRA 



CHAPTER I. 
FIRST PRINCIPLES. 

1. In Algebra figures and letters are used to represent 
numbers, instead of figures, as in Arithmetic. 

Thus, we may use x to represent the number of dollars in a man's 
business, the number of cents in the cost of an article, the number of 
miles from one place to another, the number of persons in our class, 
etc. 

In Algebra, the letter x is reasoned about and operated upon just 
the same aa the numbers which it represents are reasoned about and 
operated upon in Arithmetic. 

2. Symbols of Operation. The signs +, — , X, and -h, 
are used to deuute the algebraic operations addition, sulv 
traction, multiplication, and division, that in Arithmetic 
can actually be performed. + is read 'plus; — is read 
miniis ; X is read multiplied by; H- is read divided by. 
A dot or point is sometimes used instead of the sign X. 
Thus, a X 6 and a • h both mean that a is to be multiplied 
by h. The multiplicand is usually written before the 
multiplier. 

Dimsio-ii in Algebra is more frequently represented by 
placing the dividend as the numerator, and the divisor as 



2 ELEMENTS OF ALGEBRA. 

the denominator of a fraction. Thus, a -i- b, or - , means 

b 

that a is to be divided by b. Eead a divided by b. 

m 
Note. Do not read such expressions as — , m over n; it is meaningless. 

3. We must be careful to distinguish between arith- 
metical and algebraic operations. The former can actually 
be performed, whereas many operations in Algebra can only 
be indicated. 

Thus, suppose a man owes ^ 5 for a vest and % 20 for a coat, actual 
addition gives $25 as his total indebtedness. But if the number of 
dollars he owes for the vest be represented by m, and the number of 
dollars that he owes for the coat be represented by n, his entire debt 
can only be indicated. In order to show that the number represented 
by m is to be added to the number represented by n, we use the sign 
+ written between them ; thus, m + n. 



Exercise I. 

Eead the following algebraic expressions : 

1. 0! + 100 ; a + 10 - 2 ; & - 2 ; & - 100 + 8. 

2. a -\-b; m -\- n -{- ^ \ m + s ~ r; a — b + m. 

3. c + 2x5; c -10 X2; s-nX r-20. 

4. q + t + S X m; c + m^n — s - q; — - + c-^a—p -{- 1 - x. 

a 

Indicate by means of algebraic expressions the following: 

5. The sum of m and n. The difference between m 
and n. The sum of x, y, and a. 

6. The sum of m, n, and r diminished by t. If you had 
m cents, earned n cents, and are given r cents, and then 
spend t cents ; how many cents will you have left ? 



FIRST FRINCIPLKS. 3 

7. John has m apples, Henry has n apples, and Charles 
has b apples ; express the number of their apples. How 
many more have John and Henry than Charles ? 

8. If you buy goods for a dollars and sell them at a gain 
of b dollars, express the selling price. 

9. I buy goods for m dollars and sell them at a loss of 
71 dollars ; express my selling price. 

10. Henry had x marbles ; he gave John vi marbles, and 
Charles ii marbles. How many had he left ? 

11. I pay n cents for a reader, x cents for a history, y 
cents for a grammar, 6 cents for car-fare, and have m cents 
left ; express the number of cents that I had at first. 

12. A boy earned a dollars, then received m dollars 
from his father, n dollars from his mother ; and spent k 
dollars of what he had for books, x dollars for a coat, and 
y dollars for a sled. Express the number of dollars he had 
left. 

4. The Sign of Multiplication is generally omitted in 
Algebra, except between figures. Thus, 

bah means bXaXb, prstuz means p X r X s X t X v X z ; 
J • 3 • 4 • 5 means 2 X 3 X 4 X 5, or 120. 

Again, if the numher of gallons in a cisk of cider is represented by 
a, and the number of cents in the cost of one gallon is represented 
by m, then the number of cents in the cost of n casks is represented 
by amn. 

5. In the expression 5 + 2«o — a+ -— — : 5, 

m 2 am ^ 

2 ab, a, — , and -ttt- &re called Terms. 
n ooc 



ELEMENTS OF ALGEBRA. 



Exercise 2. 

Eead and state the meaning of the following algebraic 
expressions : 

1. 5ahx-\ ah. Result : 5 times a times h times 

c 

X, plus 7n times n divided by c, minus a times b ; etc. 

2. kl + 1-t; PQrs + ab cd + mnx y — 80. 

en b 

, ^ ^ ^klx abed a. i . i-i 

3. amnpgr — cdXo-\-- ; Zb d+ 11— r. 

, bwyz mnop 

4 bx'^+12pqrst-Q^hk-^a-\-imz. 
ab 

„ S ab d — 10 mnr + Imnr st 

5. . 

ad — 1 

6. h + '^-^+u; 2^+2^^+^^. ?±fZ^* + „ + *_±f . 

4 y u X a I 

6. It is customary to write the letters in the order of 
the alphabet. 

In a product represented by several letters and numbers, the num- 
bers are written first. Thus, 

cX&XaX5X3 is written 3 X 5 ahc ; both mean 15 a 6 c. 
Also, s X r X n X m X 25 is written 25 mnr s. 



Exercise 3. 

Write algebraic expressions for the following : 

1. The product of x, y, and z ; of m, n, and 5 ; of 3 and 
xy, of 5, a, b, and S X mn, 

2. The product of a and b divided by their sum. Their 
product divided by their difference. 



FIRST PHINCIPLES. 5 

3. The product of m, n, r, and 25 divided by the sura of 
m and n. The same product divided by the difference of 
VI and 71. 

4. A travels at the rate of 3 miles an hour ; how many 
hours will it take him to travel 30 miles ? How many 
hours to travel a miles ? To travel m n miles ? To travel 
60 a in n miles ? 

5. A man bought 18 loads of wheat, of m bushels each, 
at n cents a bushel ; how many cents in the entire cost ? 

6. In example 5, suppose that he sold the wheat at a 
gain of r cents a bushel ; how many cents did he gain ? 
How many cents in the selling price ? 

7. In example 5, suppose that he sold the wheat at a 
loss of a cents a bushel ; how many cents would he lose ? 
how many cents in the selling price ? 

8. A man bought a boxes of peaches, each containing h 
peaches, at c cents a peach ; and m baskets of grapes, each 
containing n pounds, at r cents a pound. How many cents 
did he pay for both ? 

9. A man worked n hours a day for m days, at a cents 
an hour. With the money he bought a coat for x cents ; 
how many cents had he left? 

10. One boy sold a apples at c cents each ; another sold 
n peaches at m cents each ; a third sold r peai-s at t cents 
each. How many cents did they all receive ? 

11. I buy 5 tons of coal at SIO per ton, and pay for 
it in cloth at S2 per yard ; how many yards will it take? 
I buy a tons of coal at h dollars per ton, and pay for it in 
cloth at m dollars a yard ; how many yards will it take ? 



6 ELEMENTS OF ALGEBRA. 

12. A man works n weeks at h dollars a week, and his 
son works m weeks at r dollars a week. With the money 
they pay for c cords of wood at d dollars a cord ; how many 
dollars have they left ? 

13. If 5 cords of wood cost $15, how many dollars will 
3 cords cost? If c cords cost $m, how many dollars will 
n cords cost ? 

14. A man drove 3 hours at the rate of 10 miles an 
hour ; how many hours will it take him to walk back at 
the rate of 6 miles an hour ? If he drives 3 days n hours 
each day, at the rate of t miles an liour, and 5 days m hours 
each day, at the rate of s miles an hour, how many hours 
will it take him to return over the same distance, at the 
rate of r miles an hour ? 

15. If you buy t tons of coal at the rate oi %d for n 
tons, and sell it at a loss of $ Z on each ton, how many 
dollars will you receive ? Suppose you sell at a gain of 
%h on each ton, how many dollars will you get for it ? 
Suppose you sell all of it for r dollars, and make a profit, 
how many dollars profit will you get ? 

7. Symbols of Relation. The signs =, >, and <, are 

used for the words, equals, is greater than, and is less than, 
respectively. 

Symbols of Aggregation. The signs ( ) , [ ] , { } , and , 
are used to show that the terms enclosed by them are to 
be treated ns one number. They are called parenthesis, 
bracket, hracc, and vinculum, respectively. Thus, 

(2 ft + 6) (3 X - y), [2 a + h][3x - yl {2 a + b] {3 x ~ i/], 
2a i- b X 3x — y, each shows that the number obtained })y adding 
the terms 2 a and b is to be multiplied by the result obtained by 
subtracting y from 3 x. 



FIRST PRINCIPLES. 7 

Sjrmbols of Abbreviation. The signs (of deduction) .*., 
(of reason) •.*, and (of continuation) ...., are used for the 
words, heiice or therefore, since or hecmise, and so on, 
respectively. 

8. Since 81 = 9 X 9, or written 9^ for brevity, 81 is 
called the second power of 9. Since 27 = 3 X 3 X 3, or 
written 3^ for brevity, 27 is called the third power of 3. 
Similarly a^, {m n)\ (m + n)^, are called second poioers of 
a, m n, and m -^ n\ also a^, (m n)^, {m -\- n)^, are third 
powers of a, m n, and m -\- n. cfi means a X a ; a^ means 
a X a X a \ etc. In general, ft" is called the nth power of 
a, read a 7ith power. 

9. In the expression a^ + h^c^ — 3 x"; 2, 4, 5, and ?i 
are called Exponents, h^ c^ means bxbxbxbxcxc 
xcX cXc; 4 and 5 are used for convenience to show how 
many times b and c are used as factors. 

We must be careful to keep in mind the meaning of each indicated 
operation when rea(hng an algebraic expression. Thus, the expres- 
sion 5 X* j/"^ — 2 a* 6 («' — 6*)^ + 3a^c*d"* means, five times the third 
power of X times, the second power of y^ minus two tim^s the fourth 
power of a times b times the fifth power of tlie expression in the 
parenthesis, a seventh power minus b sixth power, phis three times 
the fifth power of a times the fourth power of c times the with power 
of d. 

Exercise 4. 

Read the following : 

1. m^;3m^x^;5m^nY,'fh(M^(^iab^-hh,vL^-n^\ lOaVr^. 

2. m^n^ -f 5 a^bxi/ - 3 m^^ x^; m^n^ -'lab m n + (H'K 

3. 10 (ft 6)^0; (?n3n8) (m nf\ (a^ - „)2. (,„2 _ 3 ^^^a 

4. (m n - m3)3; 3 (v^ b {<i - }?'^- (n^ + />") (a^ - b^f- 



8 ELEMENTS OF ALGEBRA. 

5. 3 oTlf', 3 {a bf ; a^ (b^ - c^ - d'^f^ (m^ n^) (m n)\ 

6. (10 m + n"^) (10 n^ - m^f < 15 a (x - i/)^ (x + yf ; 

(1^ + 1)^ + d^ + ^7 (c5 + d^y 

b{a + h + ef ^ ^'' ^ c2 + 6^3 ' 

7. •.•a + 2:c=:6 + ^, .'.x = h — a\ {a'^ — c''f={7n?+n^f, 

... a"* - C'* =r m2 ^ ^2 . ^2 - ^-3 = 2 ^3 _ 2 ,^,2^ .-.0.^3 = ,^,^2 . 

x-\-x-^x-\'X-\- to n terms =^ nx\ a X a X a X a X 

.... to 71 factoi'S = a" ; 1 -^ x ■\- x^ + st? -\- .... 



..2 I ^, ^^ I f^ ., f^,„«o _ ^('''" ~ ^) 



1 -a? 

a + ar + a ?'^ + a ?^ + . . . . to 7«. terms = , . 

r — 1 

Write algebraic expressions for the following : 

8. The sum of m and n. The double of x. The second 
power of the sum of a and h. The second power of differ- 
ence between x and y. Five times the third power of the 
difference of x and y. 

9. The second power of the sum of x second power 
and y. The second power of the sum of x and y second 
power. The product of the fourth power of x, the third 
power of ?/, and the second power of 7n. The product of 
the first power of x and three times the nth power of y. 
The product of x second power plus y second power, and 
X second power minus y second power. 

10. The product of the sum of x second power and y 
by n a. Five n third power minus seven m n plus six a 
second power, m third power minus two times b second 
power c plus n fourth power is equal to n times y. 

11. Seven times m fourth power times n second power 
minus two times -^t seventh power times m third power 
plus three times a tliird power times h second power plus 
eight times a second power times b third power plus five 



FIRST PRINCIPLES. 9 

times a fifth power. Since a plus h equals m minus w, 
therefore the second power of a plus h is equal to the 
second power of m minus ii. 

12. Therefore, x is equal to m third power, because x 
plus three m third power is equal to two x plus two m 
third power, a plus a plus a, and so on to n minus two 
terms, equals n minus two times a. The second power 
of m plus yi, divided by m minus n is less or greater than 
m times a plus h plus c plus o? plus e. a less than 6 is 
equal to iii greater than n. 

13. A horse eats a bushels and an ox h bushels of oats 
in a week ; how many bushels will they together eat in n 
weeks ? If a man was a years old 50 years ago, how old 
will he be x years hence ? 

10. The Nomerical Value of an algebraic expression is 
the number of positive or negative units it contains, and is 
found by giving a particular value to each letter, and then 
peiforming the operations indicated. Thus, 

If a - 3, 6 = 4, x = 5, y = 6, find the numerical values of : 

,2,. 9 6x« 
25 a* 1/2 

Replacing? the letters in each expression hy the •particular values 
given for them, we have 

Process. 4 a«6« = 4 x 3^ X 4» 



= 4X9X64 
= 2304. 



9h3* 9 X 4 X 5« 
25rt»y2~25 X 3»X 62 
9 X 4 X 125 



25 X 27 X 36 

= A 
27' 

U. If one factor of a product is equal to 0, the whole 

product nmst be equal to 0, wlmtever values the other factors 



10 ELEMENTS OF ALGEBRA. , 

may have ; and it is also clear that no product can be zero 
unless one of the factors is zero. Thus, ah is zero if a is 
zero, or if b is zero; and if ab is zero, either a or 6 is zero. 
Again, if a; = 0, then a^h'^xy^ = 0, also ax{y^ + 62; + «^) 
== 0, whatever be the values of a, 7;, 3/, and z. 

Exercise 5. 

If <x = 6, & = 2, c = 1, a; = 5, ?/ — 4, find the numerical 
values of the following algebraic expressions : 

L 3c2; 72/3; 5a&; 9^?/, 8Z?3. 3-^5. ^^8.7^4. ^^^.10. 3 ^^4^ 

2. 9 6*; 2 a a; ; 3/^ , 10 x^ ; \y^\ 5 6?/; |^ r^^ ; ^^ab cxy. 

3. 3^2 ^^ 5 1^ ^ ^ ; 7 c^ ; f .«3 ; a* x^ ; 8 a^^ 2 3/* ; | acxy. 

li 7n = 2, n = 3, p = I, q = 0, r = 4, s = 6, find the 
values of: 

4 ^^. Pi,„,2^. 4^^^■^ ?ZL!!_^ ^J!^. o«»9n. 2m2g 

p; 8^'^^ r, 6 06 5 s 4 5 7776 r- 27 m'- 64 2' 
9 7?t3 ' 6 ' ' /^ ' 54 ^m ' 32 ' r» ' 3" 

Example 6. Find the value of 5 6^ + j% x y - 5 a^ - ^a^b^, 
when (1 = 2, 6 = 3, x = 5, and i/ = 10. 

Replacing the letters in the expression by the particular values 
given for them, we have 

Process. 

563 4- ^3_xy _ 5a2_ I Q^'2^3 = 5 X 33 +^3_ X 5 X 10 - 5 X 22 - I X 22 X 33 
= 5X27 + 3X5 -5X4-3x27 

= 49. 

Example 7. Find the value of mny^-\- mhi xyt + m"nh's — -j-^ , 
wlien ?7i = 5, u = 2, r = 3, s = 4, X = 0, and y = I. 



FIRST PRINCIPLES. 11 

Process. 

mny*-Hm*nxytfm%V8-^ = 5 X 2 X IHO + S-'X 2» X 3<X4- ^ ^ ^.^ 

25 X« 
= 5X2X1 + 25X8X81X4-——^ 

= 10 4- 64800 - 50 
= 64760. 

If a = 1, b = 2, c = 3, and d = 0, find the numerical 
values of the following algebraic expressions : 

8. 10 a — 4:b -{- 6 c -\- 5 d] ab + hc + ac — da, 

9. 6 ab-Scd-{- 10 a d-2b c+ 2bd; 2 be + 10 cd. 
10. a^ + Ir^-\-c^-(P; abc + 10 bed + 5 ac d + 'S abd. 

.11. a* + h^ -\- c'-d; +ob-8e -{- ad; —40ad-\-ab. 

12. 5a+3c-G6+6(/; 36co?-|-2acc?-10rt&rf. 

13. 5/>r3 + .^3 ^_ /,3_ i.25a&3c; 15 «2 ^_ jj. r* + 10 a &. 

.Cw/2/,4 .IftS 



14. c3- 8 «(/6^- 5 ftio^. j^^y. 



c3 ai« 62 c ' 



15. 125a6cc/*m + ^,-^^l^'; «» 4. ?,3 + ,3 + ,^;^. 
80 X 

V\ 2 ., ?,3 4. iL ^ 2;{c - ^^, 3 «2i3,;6^ ^ '_' _ 2^' 



/; a' 



8 , , 3 



) a 



** 2^-'^ c^ abc 

8 ^/ /•" 8<* ^'^ r^ 

19. J a f^ ;/ -\- I n — | ^^2 d^xy\ Wa^^^^ '^ ~q~ ~ ^- 

o o 

12. Negative Numbers, if a person owes a debt of ten dollars, 
and \\di» but .six dollars in money, he can pay the debt only in part. 



12 



ELEMENTS OF ALGEBRA. 



For his six dollars in money will cancel only six dollars of his debt, 
and leave him still owing four dollars; we may consider him as be- 
ing worth four dollars less than nothing. The total number of dol- 
lars that he is worth may be represented by — 4, because it will take 
four dollars in addition to the six dollars to pay the debt. If a person 
gains eight dollars and loses eleven dollars, the number of dollars in 
his net loss may be represented by — 3, because it will take three dol- 
lars in addition to the gain to balance the loss. Similarly, if he gains 
100 dollars and loses 120 dollars, the number of dollars in his net loss 
may be represented by — 20. To enable us to represent these num- 
bers, it is necessary to assume a new series of numbers, beginning at 
zero and descending in value from zero by the repetitions of the unit, 
precisely as the natural series ascends from zero. To each of these 
numbers the sign — is prefixed. The negative series of numbers is 
written thus : 



10, -9, -8, -7, 



6, 



-1, 0. 



For convenience the algebraic series of numbers is represented as 
follows : 

Scale of Numbers. We may conceive algebraic numbers 
as measuring distances from a fixed point on a straight line, 
extending indefinitely in both directions, the distances to 
the right being positive, and the distances to the left nega- 
tive. From any point on the line, measuring tovjard the 
right is positive and tovmrd the left negative. 



-^-h 






-lis 



-ilo 



-l5 



llO 



+ 115 



In the above illustration consider A the zero or starting-point on 
the scale of numbers, and the distance between any two consecutive 
numbers one unit. The distances to the right and left of A are posi- 



^ 



i 



FIRST PHINCIPLES. 13 

tive (+) and negative (— ), respectively, as indicated by the direc- 
tions of the arrows. 

To add + 9 to + 4 (read 9 and 4 in tlie positive series) ^ we start at 
4 in the positive series, count nine units in the positive direction, and 
arrive at 13 in the positive series. That is, + 4 + (+ 9) = 13. 

To add + 9 to — 4 (i-ead 9 in the positive series and 4 in the negative 
series) f we begin at 4 in the Jiegative series^ count nine units in the 
positive directioiiy and arrive at 5 in the positive series. That is, 
-4-f (+9) = -+-5. 

To add — 9 to -I- 4, we start at 4 in the positive series, count nine 
units in the negative direction, and arrive at 5 in the negative series. 
That is, + 4 + (- 9) = - 5^ 

To add — 9 to — 4, we start at 4 in the negative series, count nvie 
units toward the left, and arrive at 13 in the negative series. That 
is, - 4 + (- 9) = - 13. 

To subtract + 9 from + 4, we start at 4 in the positive series, count 
nine units in the negative direction, and arrive at 5 in the negative 
series. That is, + 4 - (+ 9) = - 5. 

To subtract + 9 from — 4, we begin at 4 in the negative series, 
count nine units in the negative direction, and arrive at 13 in the 
negative series. That is — 4 — (4- 9) = — 13. 

To subtract — 9 from + 4, we begin at 4 in the positive series, 
count nine units in the positive direction, and arrive at 13 in the posi- 
tive series. That is, + 4 - (- 9) = + 13. 

To subtract —9 from —4, we start at 4 in the negative series, count 
nine units in the positive direction, and arrive at 5 in the positive 
series. That is, - 4 - (- 9) = -|- 5. 



13. The sign + is often omitted before a number in the positive 
series. Thus, the numbers 3, 5, and 6, taken alone, mean the same 
^ (+3), (-f- 5), and (+ 6), showing that the numbers are in the 
positive series 

The sign — must always be written when a number is in the 
negative series. Thu.s, the numbers 3, 5, and 6, taken in the negative 
series, are written (- 3), ( - 5), and (- 6). 

The Algebraic Signs -f- and — mark the direction that 
the numbers following them are to take. These signs are 



14 ELEMENTS OF ALGEBRA. 

used to indicate opposition (opposite direction), also opera- 
tion. The former is called the positive, and the latter the 
negative sign. 

An Algebraic Number is one -Which is represented by an 
algebraic term vnth its sign of direction. Thus, + 3,-3, 

— a, and + 5 a are algebraic numbers. 

Absolute Value shows what place a number has in the 
positive or negative series. Thus, + 3 and — 3 have the 
same absolute value ; that is, three units. 

Absolute Numbers are those not affected by the signs -f 
or —. 

Example. The meaning of an algebraic expression, as 
3a;2+(-2a6)-[c-(-2/)], 
is explained thus : 

To 3 x^ units in the positive series add 2ab units in the negative 
series, and from their sum subtract the expression in brackets, c in the 
positive series minus y in the negative series. The signs written 
before the terms (—2 a 6), (— y), and before the bracket, indicate 
operation. The sign written before 2 ah and y, also the sign under- 
stood before 3 x^ and c, indicate opposition. 

Exercise 6. 

1. Over how many units and in what series of numbers 
would a point move in passing from 4-3 to — 8? — 10 to 
+ 1? +5 to +15? -12to-l? -lto-12? 15 to 5? 
9 to 9 ? — 5 to — 5 ? 

2. Which is the greater, or — 6 ? 3 or — 3 ? — 5 or 

— 3 ? + 10 or — 1 ? + 50 or — 50, and how many units ? 

3. How many units is + 6 greater than 0, + 3, — 3, — 6, 
and — 5 ? How many units is — 5 less than 5 ? How 
many units is a less than b ? 



FIRST PRINCIPLES. 15 

4 If a point start at (-f 3) and move three units to the 
right, then five units to the left, where is it? Express its 
distance from 0. 

5. Suppose a point start at (+ 2), and move six units 
to the right, then eleven units to the left, where is it ? 

6. Where is the point which, starting at (— 5), moved 
(— 3), then (+ 8) ? Express its distance from the starting- 
point. 

7. Suppose a point starting at + 3, move + 2, then — 7, 
then -f 5, then — 6, then + 10, then — 1-1, where would it 
be ? Express its distance from -I- 3. 

Explain the meaning of: 

8. 2 [3 6 -f (- 5 a)] - 5 [(- a) + (+ h)]. 

9. (+ 8a;) 4- [+ 3x-(+ 12 y) + (- x) - (8 y)]. 

Also the meaning of the signs + (as used or understood) 
and — . 

Explain the meaning of: 

10. 6«5A2 -f (-\-a^}fi) -f (^aH^c^) + (-aH'^) + (+aH^) 
+ 20 a2^c2 + (- aH^) + (- a^b^). 

11. aU^ ^ ^_ ^^.10) ^ (^ ^6 ^5^ ^_ (_ y a.) + ,;i3 ,^ft 

- (4- a^° m^). 

12. + (+ a^) ~{-\-h^)- (+ a^) - (+ ^). 

13. (x + 7/)2 + (a + xf - (x + 7/)2 - (a + x)^ 

Find their numerical values when a^I?^c^m = n = k 
= y = x=l. 



16 ELEMENTS OF ALGEBRA. 

Eead the following expressions, and find their numeri- 
cal values when a = 0, b = 1, c = 2, d = S, e = 4, 
n = 6, and m = 6 : 

14. c5 - (+c) - (+ n^) (+ c) - (+ d^) + a^(+ h^) (+ <^\ 

15. 3 [g + (+ 7ij\ - 5 (+ c) -h (a + 6) + 2 (+ «) ^ 6. 

16. (+c) [a + (4- ?i) + (+^) - (+6)] - (+ m) -r (+ d), 

17. [(+m) - (+rO + (+m) - (+e)(+c)] - [(+m) (+6)]. 

18. d^^{^d')^'2{+h')-^{+c')-{+c^)-^d-e^-^{-\-4:% 

If tt = 5, & = 4, c = 3, t^ = 2, and e = 1, find the numeri- 
cal values of the following expressions : 

19. (+a2)4-(+/^3)_(+c2)--(+e5); a'^c-ahl?+d^-^{a'^W'). 

If a = 1, 5 == 2, c = 3, c? = 4, c = 5, find the values of: 

26+2 3c-9 e2-l d^ 8a24.3?>2 4,2^552 c^^.^ 
e-3 6-2"^e + 3' F' «2+Zy2 + c2-62 ^2 



d>-h'' eh c ' b'^+d^-bd' e^+ed+d^' 

a^ + 4a%+6a%'^ + 4:ab^-{ -h\ 28 12 



a^ + Sa^b + Sal)^ + b^ ' o? + h^j\-c^^ d?-c^-}p^' 

24. aJ— 156-^5; — ^ ^ ; . 

a -\- b c + ft o + c 

a*-4a3(?4-6«2c2_4r?.^3_|_^4 ,^ , ,^ ^, ^^ 

25. ,4 ^xq . ...22 .z.q 7 4 ; 12e-4a-^(2aX5)-26. 
b*—4:Wc+bb^c^—4:b(f-\-c^ ^ ^ 

26. [(12e-4a)-^2«] XZ^; a^ ^ {aHU^) ^ (^c^ - a'^y 



29. 



FIRST PRINCIPLES. 17 

27. -T-i ;, , > + ^ . . , , ; (« + ^) (6 + c). 

28. (6 + c) (c + rf) + (c + rf) (rf + e) + -^_^ji^_^. 

30. 66-^-(a — c) — 3rf + a6ctZ-i-24a;c + 5f]^-Ho 
+ a X e. 

Express the following statements in algebraic symbols : 

31. To the double of a add 6. 

32. To five times x add h diminished by one. 

33. Increase h by the sum of a and x divided by y. 

34. Write x, a times. 

What is the sum oi x-\- x-^x ■\- .... written a times ? 

35. Write three consecutive numbers of which n is the 
least. 

36. Write five consecutive numbers of which m is the 
greatest 

37. Write w, a minus 1 times ; also m plus n times. 

38. Write seven consecutive numbers of which x is the 
middle one. 

39. Write a, icth power, minus y, nth power. 

40. To the double of x, increased by a divided by &, add 
the product of a, J, and c. 

41. To the product of a and h add the quotient of x di- 
vided by a, and divide their sum by y diminished by c. 

2 



18 ELEMENTS OF ALGEBRA. 

42. Write a exponent n plus the quotient of x divided 
by y, minus h times the quotient of h divided by the ex- 
pression, a exponent c plus & exponent m, is greater than h 
minus x, 

43. Write x fifth power minus h sixth power ])lus y to 
the ??ith power, divided by z to the n\\\ power, is less than 
g- tenth power. 

44. Write a to the ??-th power divided by h exponent m, 
minus x exponent n, equals a minus h, divided by the sum 
of a second power and h third power. 

45. Write c fourth power divided by a second power, 
minus the product of ^ and y, plus x .... written n times, 
equals a exponent m. 

46. X exponent m, plus the fraction, a fifth power minus 
three times a second power h third power, divided by x 
minus y, equals x minus y, added to the sum of 4 a and h 
minus m, plus 1 divided by x to the nth power. 

47. Five times the third power of a, diminished by 
three times the third power of a times the third power 
of &, and increased by two times the second power of h. 

48. Three times x exponent 2, minus twice the product 
of X exponent 3 and y, plus the third power of a. 

49. Six times the third power of x multiplied by the 
second power of y, minus a exponent 2 times the fourth 
power of h. 

50. a times the second power of n, divided by x minus 
?/, increased by six a times the expression x plus y 
minus z. 



ALGEBKAIU ADDllluN. 



19 



CHAPTER ir. 

ALGEBRAIC ADDITION. 

14. In Art. 12 it was shown that to add a positive 
miniber means to count so many units in the positive di- 
rection, and to add a negative number means to count so 
many units in the negative direction. 

In Algebraic Addition of several numbers, we count from 
the phice in the series occupied by any one of the num- 
bers, as many units as are equal to the absolute value of 
the numbers to be added and in the direction indicated 
by their signs. Thus, 

ExAMPLK I. Fiml the sum of 3 a and — 9a 

Solution. 3 a signifies a taken 3 times in the positive series, and 
— 9a aigniiies a taken 9 times in the negative series We count from 
+ 3 a, 9 a units in the negative direction, and a is tiikeii ia all 6 times 
in the negative serie.*?, or —6a. That is, 3a+ (— 9a) = - 6a. 
Similarly (-}- 9 a) + (- 3 a) = 4- 6 a. 

Example 2. Find the sum of a, 2 6, and (— 3 c). 



> + 









"N^ 




>i 


+Z^ 




A D 




c 




o 
-3C 


< 





20 ELEMENTS OF ALGEBRA. 

Explanation. Suppose these algebraic numbers to be accurately 
measured as represented on the line of numbers A C. Start at By 
then count 2 b units in the positive direction and arrive at G. Now 
count 3 c units in the negative direction, and arrive at D in the posi- 
tive series. 

Thus, (+ a) + (+ 2 6), or a + 2 6 = ^ C ; 

a -{- 2b + (- 3 c), or a + 2h - 3 c = A D. 

The sum of the algebraic numbers is equal in absolute value to 
A D in the positive series. That is, 

(+ «) + (+ 2 6) + (- 3 c) = a + 2 6 - 3 c. Hence, 

The sum of several algebraic numbers is expressed by con- 
necting them loith their proper signs. 

Notes 1. The sum of several algebraic numbers is the excess of the num- 
bers in the positive series over those in the negative series, or the excess of the 
numbers in the negative series over those in the positive series, according as 
the one or the other has the greater absolute value. Thus, in Example 1 the 
algebraic sums are —Qa and + %a. In Example 2 the algebraic sum is A D 
in the positive series. 

2. The sum of algebraic numbers is the simplest expression of their aggre- 
gate values. 

3. Algebraic addition is not always augmentation as in arithmetic. Thus, 
(+ 7) + (- 5) = 2 ; also (+ 8) + ( -ll) = - 4. 

15. A Coefficient of a term is d^ factor showing how 
many times the remainder of the term is taken. Thus, 

In the term 5 abm, 5 is the coefficient of a b m, and shows that 
a 6 m^s taken 5 times ; 5 a is the coefficient of 6 m ; 5 ab is the co- 
efficient of m. In the term 4 m (a 6 - 2 a), 4 is the coefficient of 
m (rt 6 — 2 a) ; 4 w is the coefficient of (a 6 — 2 a). 

Note. A coefficient may be numerical or literal. When no nnmerical 

coefficient is expressed, 1 is always understood to be the coefficient; as, x ; xy°. 

Like Terms are those having the same letters affected 
with the same exponents. Thus, 



ALGEBRAIC ADDITION. 



21 



2m^ujc^ </*, m"^ n jfi y*, and — 10 m* n x* y^ are like terms, as are 
also bx^^y^z* and — 3x"y*z*; but dx*y^ and 5x*i/*2* are un- 
like terms. Like terms are said to be similar. 

A Monomial or Simple Expression consists of one term; as, 
x: lU (( l>c: — 5 a^ j^. 



16. To Add Similar Monomials. 

I. When all the Terms are Positive or Negative. Add 
the numerical coefficients; to the sum, annex the common 
symbols^ and prefix the common sign. 

II. When Some of the Terms are Positive and Some are 
Negative. Add separately th£ numerical coefficients of all 
tlie positive terms and the numerical coefficients of all the 
negative terms; to the difference of these two results^ annex 
the common symbols, and prefix the sign of tlie greater sum. 



EXAMF'LE 1. Find the sum of 10 a; y*, - Sxy^ 4x?/*, 
and —\lxy^. 

Explanation. For convenience write the terms 
as shown in tlie margin. The sum of the coeffi- 
cients of the positive terms is 14, and the sum of 
the coefficients of the negative terms is 'SI. The 
difference of these is 17, and the sign of the 
greater sum is negative. Hence, the required 
sum is — nxy*. 



Uxf, 



Process. 

+ Axy^ 

- 3xy^ 

- Uxy^ 

— M xy^ 

— 17x2/* 



Example 2. Find the sum of (x -f y), 1.1 (ic + y), - 2.9 (x + y), 



.29 (x + J/) . - i (x + y), and 1 .26 (x-\-y). 

Explanation, (x+y), enclosed in parentheses, 
is treatt^l as a simple symbol. The coefficients 
of (x + 1/) are 1, l.l, 2.9, 29, i, and 1.26. The 
sum of the coefficients of the positive terms is 3.65, 
and the sum of the coefficients of the negative 
terms is 3.15. The difTerence of these is .5, and 
the sign of the greater sum is positive. Etc. 



Process 

+ C-^ + y) 
1.1 (x + y) 
+ .29(x-f-2/) 
-f 1.26 (x 4- 2/) 
-2.9 (x + y) 

- H^ + y) 

+ .5(x-fy) 



22 ELEMENTS OF ALGEBRA. 

Exercise 7. 

Find the sum of: 

1. (+ 2 a), (+ a), (+ 4a), (+ 3a), (+5 a), and (+ la). 

2. (+ 5 a a?), (+ 2 a x), (+ 6 a x), (+ a x'), and (+ a x). 

3. (+ 6 c), (+ 8 c), (+ 2 c), (+ 15 c), (+ 9 c), and (+ c). 

4. (-6a&c), (+4«&c), i+ahc), (-2abc), and (+5a&c). 

5. (-fa;2), (-|^^2)^ (-1^2)^ (-i^''), and (-x^). 

6. (+ f ^), (- I ^), (+ I ^\ (- 2 ^), (+ I x), and (+ ^). 

7. (+ 3 a3), (- 7 a3), (- 8 a«), (+ 2 a^)^ and (-11 a^). 

8. (+ 4a2?>2)^ (_ aH2), (- 7 a^h^), and (+ .5a2&2). 

9. (+7ahcd), (+ 2ahcd), (-l.labcd), and (-4.1aZ>ctQ. 

10. + (ft -^ 6'), - .01 {b + c), + .7{b + c), -10 (b + c), 
and + i(b + c), 

11. +10(,.— 2/)3, -(x'-2/)3, +m(x-yf, -2{x-y)\ 
and — 3 (ic — ?/)^. 

17. If the monomials are not all like, combine the like 
terms, and write the others, each preceded by its proper 
sign (Art. 14). 

Example 1. Find the sum of (+ 7 k), (^- 3 h if), (- 2 x), (-5 6 y% 
(+ 4 X), (-8 6 /), (+ 9 x), (+ 6 1/), (+ 1 1 x), and (- h y^). 



ALGEBRAIC ADDITION. 



23 



Ilzplanation. For convenience, write 
the expressions so that like terras shall stand 
in the same column, as in the mai-gin. 
The sum of the terms containing x is 
-f 29 X, and the sum of those containing 
b y^ is — 10 6 y"^. Hence, the result is 
-H 29x- 10 6 2/'*. 



Process. 

4- 7x4- 3 6 1/2 
- 2 X - bhy^ 
+ 4 X - 8 6 1/2 
+ 9x-f 6 1/2 
+ llx- 6y2 
+ 29 X - 10 6 1/2 



Example 2. Find the sum of -f .05 (a -|- 6), - .01 (m + n) 
4- 7 (a + 6), - 3 (m + n), - 1 1 (a + 6), and -f 10 (m -h n). 

Explanation, (a -f &) and (m + w), 
enclosed in parentheses, are treated as 
simple symbols. The sum of the like 
terms containing (a+/>) is — 3.95(a-|-ft). 
The sum of tiie like terms containing 
(m + «) is + 6.99 (m + n). 



Process. 

-f .05(0 + 6)- .01(m + «) 

+ 7 (a + 6)- 3(m4-w) 

- 11 ((! + ?>)+ 10(m + yi) 

— 3.95 (a + 6) -f 6.99 (m + n) 



Exercise 8. 
Find the sum of : 

1. (+ .3 X), (+ .5 y\ (+ .01 .>), (+ 3 y\ and (- 7 x). 

2. (4- I a), (- J a J), (+ § a), (-h ^ rr ft), and (- | ..)• 

3. (+ 5 (-2 ./2), (- 2 a«./;), (- 2 c2.r2), (+10 a^ x), (+ 8 c2 x^), 
(-4a«x), (-4c2.x'2), and (-f 4a8a?). 

4. (+ I .^), (- J a 5), (+ V •^'). (+ 1^5 ^^ ^'). (+ A^^)> 
(+^^'). (-HJ«^). (-i^^), {-i^^''). and (+§«6). 

5. 7a, - :^x-yl 8a, .3 (x-?/), .03Cr - y), and -.la. 



18. A Polynomial <>i Compound Expression consists of two 
or more terms. 



24 ELEMENTS OF ALGEBRA. 

Example. Find the sum oi 8ax— .ly + 5, .7ax + y — am — 9f 
and - .3ax- 1.02 y + 5p- .3. 

Process. Sax — .1 y +5 

.7 ax + y — am — 9 

-.Zax- l.02y ~ .3-t-5p 

8Aax — .12 y — am — 4.3 + b p Hence, in general, 

To Add Polynomials. Write the expressions so tliat like 
terms shall stand in the same column. Find the smn of the 
terms in each column, and connect the results with their 
proper signs. 

A polynomial may be regarded as the sum of its monomial terms. 
Thus, the sum of the terms (-f <?), (— h), and (—3 c) is « — ft — 3 c. 
Hence, the sum of two or more polynomials whose terms are all 
unlike is expressed b}'^ writing their terms ivith their respective signs. 
Thus, the sum of a — b, c — d, and m. + n — x is a — b + c — d 
+ m + n — x. 

Exercise 9. 

Find the sum of : • 

1. 2x-{-y, bx+3y, — 3x — 2y, and — 4 a^ + Sy. 

2. 5x + Sy+3a, —7x + 4:y — 8 a, and 2 x — 3 y. 

3. 3 6-3^, 2 c - 2 ^, 3c -71), and 4:h-2c + 3x. 

4. 14a + x, I3h-y, -11a + 2 y, and ic - 2 a — 12 &. 

5. ax — 4:mn + hd, hd — a x — 3 mn, 7 m n — 3 ax 
+ 3hd, and 5mn — 30 ax — 9hd. 

6. a — h, 21) — c, 2 c — d, 2d — 3(f-\-n, and m — n + x. 

7. a b c -\- 3 a b m — 5 c m, 3cm+ 11 abm-\-9abc, 
90abm — 21cm — 31 abc, and 3cm — 51 abm-\- 13 abc. 

8. T/i + n + p, m — n—p, m — n+ p, and m -\-7i—p. 



ALGEBRAIC ADDITION. 25 

and — a + 6 + f 4- rf. 

10. 1.25 a 6 + 1.1 c + 99 h, ami 3 ah ■{■ 2.2 c -\- 1.01 i. 

11 ia^iah + .QV', la - ^^^ « ^ " 1^ ^'. I ^' + i^^2> 
f .6 ^, and .1 a - 1.01 « h 

12. J ?/|2 - .2 wi -f J, .1 wi2 + .01 m - 2, m^ _|. 3 j ^^^ ^^ _ ^^ 
and J m — 5S m 71 — 1 J w^ — 2|. 

13. xy — ac, Sxi/ — 9a(\ —7xy + 5ac+lMcd, 
4 xy -^ 6 fl c — .09 f rf, and — x y — 2 a c + c d. 

14. .5 r(3 - 2 r|2 h - I b\ lu^b- .75 a l^ + 2 b^ and 

15. 3 (m - nf + .3 (a: + ?/)^ .4 (m - 7/)3 - .2 (x + y)\ 
.7(m-n)8-3.03(a; + i/)^ and 5.1 (m- 7^)3 - M 1 (.aH-.#. ' 

16. oa^V^-^a^b^-\-x^y-^xf, 4a^b^-7 a^ b^-'Axf 
+ (jx^y, 3a^l^-^Sa^b^-3a^y+5x y\ and 2 a^lfi 
^a^JJi ^ 3 a^ y — 'S X if. 

17. |aa.-2-§/^2 4. |.,3y^. j8^ 3^.x^+ |.r^?/+ 7.5^2 
+ J fcs 2 ^ .^2 ^ 3 ,3 ,j _ 1 ,,2 _ 1 2,8^ j^.^a Jj. a x^-V\x^ y 

18. ac^ ^ \a}?' -V In"^ - \a^b - \abc\ \a^ c, \nn 
j^lb^^laV^^ \b<?-\-2ahr + \\\?c, and 1.1 ^rc- 1 2ac2 
+ i^^c- |/>6-2l l..Sc3+ 1.23 r/ 6c. 

19. 3.1«8-4.2a?2+1.2a; + 1.7, 2.22 a,-3- 1.2 a^ + 3.33 x- 

- 10.09, 2 a^ + 7 a^ - 2 .r + 1, 3 ./;8 + 1.22 a^» + 12.12 

- 1.33 a;, and 11.1 1 j? + 5.55 x^ - 0.2 u.- + 3.77. 

20. «H2c3 + «2^^,2 ^ 3 ^^2/^3 ,5 13 ,,3^2,;3 _ 1 4 ,,2^,3^8 

+ 1.5 a2 63 c2, 1.5 «2 63 c2 - 1.9 ^3 }?• c^ ^ 1 3 ,,2 ^3 ,5 and 
1.7 a3 62(^3 _ 1.2^(2^^5 4. 101 a^ftSca. 



26 ELEMENTS OF ALGEBRA. 

21. 2c'"+.la" + 3&c, .5c'" + 3.9ft"+2.02&c, c'" + 2.09a'» 
-|&c, and i& + f Z^c- 3.03. 

22. a & — a:' + |- a??/, .1 X -{- .01 a ic — 2.02 a 6, ^ ax 
■\- ^xy — ^ah, 6ic ~ 1.01 ax + .la;, and ■J'^' 2/ + j a x 
-\xy, 

23. 3 ^ + .ly- 7.01 ^ + 6.01 ?/ + .2 a + 3 ^ - 1.5 c; 
-.8^ + 9.01a + 3.03?/ + ^-4.04?/- 2.01 2;+ 2.2 a -y. 

24. 3 a 6 4- 9 - ic2 ?y, a;3 3/ + 3 a; 2/ + 5, 6 a;^/^ + 4 a;2y 

— 3 icy, 10 o^y + 1 + 3 a; 2/2, and 17 — 3 x^y — 2 a;^y. 

25. .5 {m - 3xy, - | (m - Sx)\ .75 (m - 3 x^, and 

— 1.25 (>yi - 3 xy. 

26. ^2 + Z;4 4. c3, _ 4 tt2 - 5 c^ 8 a2 _ 7 &4 + 10 c3, and 
6 &4 - 6 c3. 

27. 3a2-4a& + &2+ 2a + 3&-7, 2^2-4^2 + 3a 
-56+8, 10 ab+ Sb^ + 9 b, and 5 ^2 _ 6 a & + 3 ^2 

+ 7 a - 7 & + 11. 

28. x^ — 4:a^y + 6x^y'^ - 4:Xy^ + /, 4:S(^y — 12x^y^ 
+ 12 xy^ — Ay\ 6 x^y^—12xy^ + 6 y^, and 4:xf — 4:y\ 

29. a3+ «62+ ac^-a^b-abc-a^c, a?b ^- b^ + bc^ 

— ab"^ — b'^c — ab c, and a^ c + b"^ c + c^ — a b c — b c^ 

— a c2. 

30. 5 a^ - 2 a^b + 9 ab^ + 17 b^, -2 a^ + 5 a^ b 
-4:ab^-12b^ b^-4:ab^-5 a'^b-a^ and 2 ^2 6 
-2 a^-Ob^-ab"^. 

31. a;»« — ?/"+3< 2ic'"-3/-a, and a;'» + 42/"— a''. 



ALGEBRAIC SUBTRACTION. 



27 



CHAPTER III. 
ALGEBRAIC SUBTRACTION. 

19. In Art. 12 it was shown that to subtract a positive 
imuiber means to count so many units in the negative di- 
rection, and to subtract a negative number means to count 
so many units in the positive direction. Hence, the addi- 
tion of a positive number produces the same result as the 
subtraction of a negative number having the same absolute 
value. 

Thus, 4-3+ (+6) = + 3 + 6 = 9. +3- (-6) = + 3 + 6 = 9. 

Also, the subtraction of a positive number produces the 
same result as the addition of a negative number having 
the same absolute value. 

Thus, + 4-(+6) = f4-6 = -2. +4+(-6) = +4-6 = -2. 

We observe that the subtraction of one number from 
iinother produces the same result as counting or measuring 
from the place occupied by the subtrahend to the place 
occupied by the minuend. Thus, 

Subtract — h from + a ; also + a from — 6. 



-a^ 



-<r 



-b 

a-b 



+b 



a<r 






IB 



D 



28 ELEMENTS OF ALGEBRA. 

Explanation. Suppose the algebraic numbers to be accurately 
measured on the line of numbers C D. We start at B in the positive 
series, count b units in the positive direction, and arrive at D ; and 
the distance from A (0) to Z> is equal in absolute value to A D in the 
positive direction. But in counting from ^ to Z> the absolute value is 
the same as the absolute value in counting from C (the subtrahend) 
to B (the minuend), and we have counted in the direction opposite to 
that indicated by the sign of the subtrahend. Thus, 

C B = -h {+b)-\- (+a) = a + b. That is, 
(+ a) -(-b) = a + b. 

Subtracting + a from — b gives the same result as counting from a 
in the positive series to b in the negative series, and the distance from 
5 to C is equal in absolute value to B C in the negative direction. 
Thus, 

BC = + (-a) + (~b) = -a-b. That is, 

(— b) — (-{- a) = —b — a, OT — a — b. Hence, 
Algebraic Subtraction is the operation of finding the dif- 
ference from the subtrahend to the minuend. 

To subtract — 5 a from + 2 a is the operation of finding hoio far 
and in ivhat direction we must go to pass from 5 a in the negative 
series to 2 a in the positive series, and is found, by counting from 
— 5 a to + 2 a, to be 7 a units in the positive direction. That is, 
+ 2a - (- 5a) = + 7a. 

To subtract + 5 a from - 2 a, we count from 5 a in the positive 
series to 2 a in the negative series and pass over 7 a units in the 
negative direction. That is, 

- 2 a - (4- 5 a) = - 7 a. 

These differences may be found by changing the signs of the sub-' 
trahend and proceeding as in addition, as shown by a comparison of 
results. Thus, 

Minuend. Subtrahend. By Addition. 

+ 2a-(-5a) = + 7a >| j^+2rt + (-t-5«)=4-7a. 

-2a- (+5a)=-7a ' I - 2 a -f (- 5 a) = - 7 a. 

+ a - (- &) = + a + 6 f ''''' ) + a + (-{- b) = a + b. 

- b-(+ a) =-a-b ) \^- 6+(-a)=-a-6. 



ALGEBRAIC SUBTRACTION. 29 

Hence, in general, 

To Subtract one Algebraic Number from another. Change 
the sign of the subtrahend, and add the result to the minuend. 

Notes: 1. Algebraic subtraction considered as an operation is not distinct 
from addition ; for it is equivalent to the algebraic addition of a number with 
the opposite algebraic sign. It includes not only distance but direction, and 
direction depends upon the sign of the subtrahend and which number is consid- 
ered the minuend. 

2. Algebraic subtraction is not in all cases diminution. Thus, 
8 ~ (- 2) - 10 ; also 2 - (- 8) - 10. 

E.xAMPLE 1 . Subtract + 3 o* 6 c w* from -f 10 a* 6 c 7/1^. 

Solution. Changing the sign of the subtrahend, and proceeding 
as in adtlition, we have + 10 a* 6 c m^ -f (— 3 a* 6 c m*) = + 7 a^bcm^. 

Example 2. Subtract + 27 (x^ - i/«)» from 13 (x^ - i/)». 

Solution. Treating (t* — y^y as a simple symbol, changing the 
sign of the subtrahend and proceeding as in addition, we have 

13 (x2 - 7/)« + [- 27 (x^ - !/«)»] = - 14 (z" - y«)». 

Exercise 10. 
From : 

1. +OaHc take —aHc, —Ual^xy^ take +19 al^xi/. 

2. +:tVtake-./.y; +99 mnp^rst^^ take ■}-99mnphst^^. 

3. —10 axy take —axy\ x'^ take —he. 
From the sura of: 

4. — 11 a.-^, + .5 d^, and + 1.25 J" take -f 5.5 0^. 

5. ah^, — Sabc^, and + .3 ah(^ take the sum of 
-abc?^, + SMabc^ and - IMabc^. 

6. lOS mnp^^, —lO.Smnp^^, and +vinp^^ take the 
sum of -- 10 771 np^^, + 33 mnp^^, and — 108.1 m np^^ 



30 ELEMENTS OF ALGEBRA. 

7. 5 {x + y), — 2{x-\- y), and + {x-\-y)y take the sum of 

- {x + v/), + 6 {x + 2/), and - 2.5 {x + ?/). 

Find the aggregate value of: 

8. + 17 a x^ -(-5a a:-3) + (- 24 a ^'S) - (+ a j:^). 

9. + 19 a ^ ?/2 + (+ a ;2; 7/2) — (— 5 a x ?/). 

10. + x-2y + (- x^y) - (+ x^y) - (- ^2y) + (_ 1 .^^2^/) 

- (+ 3 x^y) - (- 10a;2y) + {-bx^y). ' 

11. + n« + ^)' - [- -1 (« + ^)'] + [-(« + ^)'] 

- [+ I («■ + ^)'] + 10 (a + ?^)2. 

12. I .- + (+ 1 X-) + (- .1 X-) - (+ ,2^- ^") + (- 1 x^ 

+ (+ 1. ix'')-{-2>ix'y 

20. Example L Subtract 2 ah + f) a^ if - U a^ - 1 y^ from 
15 a3 - 8 1/ + 23 a3 t/S. 

Process. 

Minuend ] 5 a^ _ 8 ?/3 + 23 a^ i/S 

Subtrahend, with signs changed + 14 a* + 7 ?/^ — 5 a^ ?/^ — 3 a 6 
Difference 29 a^ - ^/^ + 18 a^ ^/S - 3 a 6 

Example 2. Subtract 3 a; ?/ + n - 5 a^ 6 + 5 jo'' from 5 ic i/^ - 3 a^ 5 

+ 3 771. 

Process. 

Minuend 5 xy^ - 3 a'^h -\- Sm 

Subtrahend, with signs changed -3a: 7/^+5 a- 6 — n — b p^ 

Difference 2xy^-\-2a^h+'3m — n — 5p^ 

Hence, iu general, 

To Subtract one Polynomial from another. Change the 
algebraic sign of every term in the subtrahend, and add the 
result to the minuend. 

Note. It is not necessary that the signs of the subtrahend be actually 
changed, we may conceive them to be changed. 



ALGEBRAIC SUBTRACTION. 31 

Exercise 11. 
Subtract : 

1. 5x~3y-f22 from 3 x -\- y — z. 

2. X — y •{■ z from — x — 2 ij — 3 2. 

3. a — 6 4- 20 6' from — a — h •{• 10 c. 

4. J^ — iy — i^ from \x -\- y + z. 

5. lx-\y-\-^zhom-lx+t^y-^^z, 

6. J^-iy-i from -Jx--Jy+ J. 

7. a3 _ 4 a2 J ^ 2 i c + 5 from 3 a^ - «U - 5 c - 5. 

8. a^y — Z ah X -\- 2 X y^ — \ from S x^y — abx-^ 2 xy^. 

9. ahcxy -\-2ahy — 4ihx from 2abcxy — ab y -\- bx — 'S. 

10. acy — bxy + aJc — 1 from bxy — 4:acy — abc -fa. 

11. .4a:*-.3a:3^.2ic2-7.1a; + 9.9 from a;^ - 2.10 a:^ 
+ .2ar»-:.07a.' + .9 

12. 1.2 /-s - 1 4- X + 1.1 a;4 + 1.7 a^ + a from \-x 
-.l.r* + .2.'/^-.3A'8 4-a. 

13. \ vin^ ^In^- yi^ + I w2 from | /7i3 - J r/i 7i2 - ;,2 

14. .125 m3 - M^ ,n vi2 _ .33^ ^ ^2 _ (5o| ^3 _^ ^. |Vom 

g Wl^ — I Wl n^ — I ?//,2 7J, -I- J 7^3 

15. J ,;i2 _ I y _ I ^ 4. J ^ |.^Qj^ ^ W2- ^ y 4- f ?l - i .^. 

16. a^hc + xf -Ic from 3^ a2ftc + 2Jaj?/"- 4f c. 

17. lOc — a-h -^ b(l-\- 6rt-15c4- 3(Z from 25«-5 
— 5c4-8r? — 20rt. 

18. af" — 3 a:" if — if from 4 x*" + .c* y" _ af . 



32 ELEMENTS OF ALGEBRA. 

19. — .9 a"* x^ - .3 a 63 a? + .6 + .03 IT c x^ from .9 a'^x^ 
+ 1.3 -2ah^x-\- AlTcx^ 

20. From the sum of lo?-\h^ +\ c\ J cv^ - 3^ c\ 
and 2| 63 - I a2 _ i| c^ take ^i a? - ^\ 63-4 c\ 

21. From the sum of 3 ic3 _ ^2 ^ _ ^^^ 7 ^3 _ il ^2;3 
-f- llj y^ oc, and 11 ;;c2;3 _ gi ^3 _ 2 1/2 a; take a;2;3 _ 25 ^/^j; 
4- 1 ^3. 

22. From of + y"" take the sum of 11 ^-^ + y^ — z, 

— 6 ^" — 5 /* — 3 ;2, and - 5 ^" + 3 2/"* + 4 z. 

23. Add the sum of S^y- .3 ^/^ and 5-3y + 2.7 2/3 
to the difference obtained by subtracting 3 + l^y^ — .5y 
from 1—2/3. 

Queries. Why change the signs of the subtrahend in subtracting ? 
Wliy add the subtrahend, with signs changed, to the minuend ? 
Does the use of the signs + and — in Algebra differ from their use in 
Arithmetic ? How ? 

Miscellaneous Exercise 12, 

1. From m^ — n — 1 take the sum of 2 n — 3 + 2 ?^3 
and 3 ??i3 — 4: + 5 'n? — n. 

2. From the sum of 1 - 8.8 y + .9 a^ and 1.1 x^ + S x^ 

— .2 2/ — 1 subtract 2 x^ — x^ + 5 y. 

3. Take x^ + x — 1 from 2 x^, and add the result to 
-2a^-x^-x + l. 

4. Take a^ — h^ from ah — h^, and add the remainder 
to the sum of a6 - ^2 - 3 6^ and ^2 + 2 62 

5. To the sum of m + 7i — 3 /? + 5 and 2 m + 3 ?i + 5^ 

— 3 add the sum of m — in — 7 p and 5 ^ — 6 ??i — 2. 



ALGKBRAIC SUBTRACTION. 33 

6. Take 3 a-^"* - 2 x-2" f^ - y^-^ from 3 f/—' + 2 x^'^y^ 

7. Take 2x^y^-*6z^-\-2y^-V z-1 from a;iy*+2;--4yi 

8. Take .8 h^ y^ - A a? x^ + .3 rf from .4 a? x^ + .3 c 

- 1.2?)tyi 

9. Take f xt-f a;*y*H- 33j?/§ from f j:;^ + |-^*?/* + Jyi 

10. From the sum of .7 c y^ — .4 a x -f .5 h, .04 6 — | cy» 
+ i w,-|a.r+ Jcy§-|, and -yi a x- .23 6- .8 m + .3 
take the sum of .55 a ic + J /?i + ^^ and .33 in — 1.1 cy^ 
+ .67 ?«. 

11. Find the sum of «"• - 7 6* -h cp and | 6" + | a"*, 
and subtract the result from cp — 4:n. 

12. From a* - 2 c" - af* take the sum of J «"• - ^ i" 

- X' and i a"* + J 6" — y" — af. 

13. From - a*-6*-c'-^« take the sum of Jft'"+ §6* 

- I C, ^5 a"* - \l c^, and V ^^ + ^'■ 

14. From 3 (a^ - ^y - (o^-a + ff take § (^ + 2^)* 
- c^ + 3J («^ - ^)*. 

15. From unity take 3 a^ — 3 a + 1, and add 5 a^ — 3 a 
to the result. 

16. Add 3 ^- - 7 a:" + 1 and 3 a;8- -f ^ - 3, and 
diminish the result by x^"* — 2. 

17. From zero subtract I a^ — ^ x -\- 2. 

18. From .3 m' - 1 + J w take 5 n^ - 2.7 m^ - J n, 
tlien take the difference from zero, and add this last result 
to - 5 n^ 4- 3.3^ m^ -f n. 



34 ELEMENTS OF ALGEBRA. 

19. What expression must be subtracted from 10 y'^ -]- y 

- 1 to leave 3 y^ - 17 y + S ^. 

20. What expression must be subtracted from a — o x 
+ y to leave 2 a — o x -\- yl 

21. From what expression must a^—bah — lhc be 
subtracted to give a remainder b a? -\- 3 ah -\- 1 h c1 

22 From what expression must a^ h^ — b^ c^ + 6 a"" c** 
be subtracted to leave a remainder b^ c^ — 6 aJ^ c" ? 

2*3. To what expression must | ft^ + 2\a — 1^ a^ — 3 
be added so as to make 2\ a^ — 2\ a -\- 3\ a^ -\- ^ ^ 

24. To what expression must b x y — lib c — 1 mn be 
added to produce zero ? 

25. What expression must be added to 3 ^" — 3 ^""^ + 2 
to produce ^" + x^"~^ — 6 ? 

26. What expression must be added to m a?"* — ^" + 2 
to produce m a?'" — 2 ? 

27. From the sum of .6 {x + y)^ + .3 a" + ^'", | a** x"^ 

— c^ ~ I (« + ?/)2, and I (^ + 2/)^ ~ I ^" ^''"j take the sum 
of .3 (x + i/)^ - I a" ;:c^, I «" ic'" - 6.5 (ic + y)^ + c^ and 
ro (^ + .?/)^ + 3.3 a" ^"^ — 3. 

Algebraic Subtraction may be defined as the operation of 
finding a number which added to a given number, will 
produce a given sum. The sum is now called the min- 
uendy the given number is the subtrahend^ and the required 
number is the difference. 



ALGEBRAIC MULTIPLICATION. 35 

CHAPTEK IV. 
ALGEBRAIC MULTIPLICATION. 

21. Evidently dm x 6n = 5 x (^ x m x n = SOmn. 
Hence, in Algebra, the product is tlie same in whatever 
order the factors are written. 

a X a X a X a or aaaa is written a*, and shows that 
a is taken four times as a factor. aXaXaXaxaov 
aaaaa is written a^, and shows that a is taken five times 

as a factor, a X a X a X ton factoi*s, or a aa to ?i 

factors is written a", and shows that a is taken n times as a 
factor. Hence, 

An Integral Exponent shows how many times a number 
or term is taken as a factor. 

a^ is read a second power, or a exponent two, or a square. 
a' is read a third power, or a exponent three, or a cube. 
Hence, 

A Power is the product of two or more equal factors. 
The degree of the power is indicated by an exponent. 

a? = a a a, 
and a^ = a aaaa a . 

Hence, a^ X a^ = a a a a a a X aaa 
= a» 
a" = a a a a .... to n factors, 
and a*" = a a a a .... to m factors. 

Multiplying the second expression by the first, we have, 

rt"* X a* = aaa to m factoi-s X a a a .... to n factors 

= a a a .... to (m + ?i) factors 

= a*"*"". In which m and n are a?iy numbers 



36 ELEMENTS OF ALGEBRA. 

whatever. Similarly for the product of more than two 
powers of a factor. Hence, 

The i^owers of a number are multiplied hy adding the 
exponents. 

If the multiplicand and multiplier consist of powers of 
different factors, we use a similar process. Thus, 

3m^ X 2 111^11? X 5 m'^n^ = o X 2 X 5m 771 m m m mmmnim 

X n n n n n 

a"fe"* X a^lf =aa a .... to n factors x aa a .... to p factors 

Xbbb to m factors Xbbb .... to r factors 

= aaa .... to {n-\-p) factors x bbb .... to (m + r) 

factors 
=: a""*"^ 5'" '^'■. Hence, in general. 

To Find the Product of Two or more Monomials. To the 

product of the numerical coefficients annex the factors, each 
taken with an exponent equal to the sum of the exponents 
of that factor. 

Notes: 1. When no exponent is written, the exponent is 1. Thus, a is 
the same as ai, & as fti. 

The exponent is used to save repetition. 

2. We read a^, a square, and a*, a cube, because if a represents the number 
of units of length in the side of a sqiiare, and the edge of a cube, then ffl2 and 
a3 will represent the number of units in the surface and volume of the square 
and cube, respectively. 

Illustrations. 

11 mi9 X 10 m}^ =11 X 10 mi^ + lo = no ni^^. 

3a'^bcmX2ah'^cmX 5abc^m^ = '3 X 2 X 5a2+i+i6i+2+ici+i+«mi+i+2 

= 30 «4 ^i c* m\ 
3a^fe*c3 X 4rt X 6'cJ = 3 X 4 a' + ift^ + J c3 + ^ = I2a^bc*^\ 
S'x^i/" X 2^x-3 X x^y"" = 25 + ^x2-3+57/" + " = 2 x*i/2". 



ALGEBRAIC MULTIPLICATION. 37 

Exercise 13. 

Find the product of : 

1. 0? and 7 a?^ ; 3 a a; and ^ c^t^ \ a? ha? and 2 a^ W a?. 

2. ^xyz^^ and Is^t/^mn; ^abcdm^7i^ and ^a^lf^c^d^mn. 

3. |^/2^.3yaud |a6366y" + '; 3a3a:«3/7andf ai«^a:8 3^*2. 

4. 3 a a:^y^ and 10 a^^xy^^ ; J x"* if and | rr* y^. 

6. 3aftca;^V and f rt-^j^c^^^ \a'hh^xy and | a Z>iOc V^^"*. 

6. f a'-fc-af / and .2 a^ h^ 2^ f , a? f and a^^i/S. 

7. a'-iz-and tt-6'"; a-^-^V^^and 5.7a:-i3/-i. 

8. ahx]/^ and a?-}?x^y\ a"'+'i>'* + '* and a'"-''/;"-'". 

9. .55a;-' + '//-'' + ' and .5 a;''+«/-''; .3 a2-"';j;8- n ^j^^ 
a* 2:"; 5a-H".r^ and ^oa\hx'^\ 

10. 2«^2:, a-?/, «2//, a^j^i/, and a ^. 

11. a*-, fc", 3c', a', 5', c*", and c?'. 

12. 2c'"</, «c", r/22:", a"* a;*", and ci 

13. ai VI x^, a^ ni x^, a m x y, and 2 a^ 71^ x^ if. 

14. a^z^, «^?/^, ala;"i, a^y~^, 5fa"la;^, and 5^2:^yV 

15. fa^ma:", jTnta:*, .Sarr"*. 5.1 w', and a-^'x-', 

16. 3y*, a-^y'^z^, (^ b"", a^b', f «^.7/", and ^a^J"'^-. 

17. (a 4- ft), 5 (« 4- 6)2, 3 (« + 6)^ ^ (^ + &)^ and (a + bf. 

18. (a + 6) (c + (if, {a + bf, 3 (c + df, and (a + 6)7 (c + rf)2. 

19. 3 (a + 6)- (a: - y)-, J (a + 6)«, and | (a: - yf. 



38 ELEMENTS OF ALGEBRA. 

22. Algebraic Multiplication is the operation of adding 
as many numbers, each equal to the multiplicand, as there 
are units in a positive multiplier ; it is also the operation 
of subtracting as many numbers, each equal to the multi- 
plicand, as there are units in a negative multiplier. Hence, 

The multiplier shows that the multiplicand is taken so 
many times to he oAded, or so many times to he suhtracted. 

Thus, 
(+ 6) X + 4 = (+ 6) + (+ 6) + (+ 6) + (-f 6) = + (+ 24) = -f 24 
(_ 6) X + 4 = (- 6) + (- 6) + (- 6) + (- 6) = + (- 24) = - 24 
(+ 6) X - 4 = - (+ 6) - (4 6) - (+ 6) - (+ 6) = - (+ 24) = - 24 
(-6) X-4= -(-6)-(-6)-(-6)-(-6) = -(-24)=:-f 24. 

■ The sign of the multiphcand (6) shows that the product (24) ii^ in 
the positive and negative series of numbers, respectively; and the 
sign of the multipl ier (4) shows that the first two products are to be 
added and the last two are to be subtracted. Hence, 

The sign of the multi-plicand shows what series of numbers 
the product is in, and the sign of the multiplier shows what 
is to he done ivith the product. 

Law of Signs. The product of two factors is positive when 
the factors have like signs, and negative when they have un- 
like signs. 

Since 

-2x-2 = +4; -2x-2x-3=:-h4X-^3==-12; 

-2x-2x-3x-4--12x-4 = + 48; 

_2x-2x-3x-4x-5 = + 48x-5 = -240; 

etc. Hence, 

The product of an even number of negative factors is posi- 
tive; of an odd number, negative. 



ALGEBRAIC MULTIPLICATION. 

The change of signs may be illustrated as follows 



39 



> + 



ax-Sr^ 


















K. 


ox+y 






»^y»w.s 






ax-3 






V 


QX-^3 


VCATT 








ClX-2 


< 






ax+i 








Q X-/ 


^ 




ax+/ 








^ 


A * 






k 




-QX*Jl 


-ax+/ 


. 


, 


-ax- A 


-fly-3 


-CtX-i' 














^ 


7 ... - 






-ffX+J^: 






•CTK^S^ 


k ■ 








-ax-s 










-| 















-< z^ 

Let the measuring unit be represented by a. 

From A (o), the starting-point on the scale, measure toward the 
right and left. The products of + a and — a by the factors from -f- 5 
to — 5 are : 

aX-l-5, aX+4, ax+3, rtX4-2, ax + 1; 
a X - 1, a X - 2, a X - 3, rt X - 4, a X - 5 ; 
-aX+5, -rtX4-4, -ax+3, -ax+2, -aX + 1; 
-aX-1, -aX-2, -aX-3, -ax- 4, -aX-5; 

respectively. The directions taken by the products are shown in the 

figure. 

ninstrationB. 

x«y» X -x*z X -^j/z^ X -^xz^ X - 4 yz^ = -f- f X ^ X 4x«j/*2»o 

X^y-* X - fx-y-Z X - yz-'- X - X~^'* = ~ ajm + n-Sny-n + n+ljl-r 

Exercise 14. 
Find the product of : 

1. 5 a, — 3 /), 7 c, — 2 a*, — 11 a^, and a; a^x, —ay^, 
a a^, and — xy. 

2. ahx, —ay^, —a X, and a^a?\ —al^, —hc^, —cd?, 
— a, — a^, — a^, and — 5 a* 



40 ELEMENTS OF ALGEBRA. 

3. — a, he, — 1, ^, 1^ a^, ^x y, and 75 a; ax, ex, 

— m X, — 2/**, and .3 ?/i. 

4. ^abc, —d, ax, —1, and ^axyz; a'^af, af^y*", a'^V, 
and a b. 

5. —a^x, 3x, ah^, ay, az, and axyvw; axy, —^a^V, 
and - SJa^&'a:"*?/". 

6. - a'^hc, 2 h'^cd^ - .5 a^ccl^, - f^ a-^H-^^c^^d'^^, 
and a h^ d*. 

7. -1, a- 3, a^7^ a 0^-5, aio^-3^ a-H-^a^, and -Mr/2 

8. aaP, — a\ — 1, .3 a x, and — a^^/^; ^^^.^ _ ^|^ ^|, 
and — a^ a^i. 

9. — my, mx, — mn, — xy, and it* 3/3; 3 aa Jt and 

— .7 «i Z>i 

10. a", a^", a^", a^**, and a^". Express the result in 
two ways. 

11. 2^ij-'x-^, mx'^y'^', -3" ^"2:-!, and -2-^po(^f. 

12. 32", -23« X 3'^^^ 32«, - 23" X 3*«, S^" x 2«, and 

— 26" X 3« + i. 

23. Example. Multiply a + & by m ; also a — fe by m. 

The symbol (a + h) m means that m is to be taken (a + V) times. 
Hence, 

Process. 

(a + h)m — m -\- m ■\- m + taken a + h times 

= {m+m+m+ — taken a times) 4-(7w + m + m+ — takenft 

times) 
~ am -\-l)m. (1) 



ALGEBRAIC MULTIPLICATION. 4l 

Also, 
(a — h)m = m-\-mi-in-h .... taken a—b times 

= (m + 7» + m-f- taken a times) — (m + m+m+.... taken K 

times) 
=:(rnX a)-(mX b) 
— am — bm. (2) 

Similarly, (a + ft — c) m = a m -f- 6 m — c m. 
These results are obtained by multiplying each term of the multi- 
plicand separately by the mnltiplier. Hence, in general, 

To Multiply a Polynomial by a Monomial. M%dtiply each 
term of (lie muUiplicaiul by Ute multiplier ^ and add the 
resiUts. 

Exercise 15. 

Multiply : 

1. hc-^-ac-ab hy abc; S aH^ - ^ hh^ - ^ c^ hy -f^a^b^^. 

2. 5a^-b'-2c^ by a^b^c^^; .6 3^- .5 2^y^- .32^y^ 

- .2a^ by .2x^f. 

3. j W.2 — ^mii -{- ^n^ by ^mn; x — y — ^x^y^ by xy. 

4. fa-^j&2_^^^a62 by|«Z^2. a' - a^lr^-ab hy ah^. 

5. 6a2a3- .5a^b^x^^ -\- .2h^2^ by ^ ab 3^; pxT-qx* 

— r by p2^ r. 

6. 3a'"-»-2 6"-H4a'"6" by a&2; .4a— " 5'''-Ja-*'6' 
+ ft3^ by I a-^-^'ft'. 

7. a*--3a'"i—4-&'* by a'-J'^"; 2^2:^ - 2iyi + 2^a;iyi 
by 2^x^yi 

8. a?-a2jf-an + 5iby aU^; a;*-2rty^4-a:^i^t-.6y* 
by xhj~^. 



42 ELEMENTS OF ALGEBRA. 

Find the product of : 

9. x^y^-4: x^f+ 4/, xh/, and -2i/; m''^ - 2m^^7V'' 
+ n^\ m~^, n-\ and in" n\ 

10. ^^26-4 + 1^6-32: + 1 62 2:2^ laV^, %b-^x, and ^aH^a^. 

11. a;3 — 7/F a;3, 2/3^ and —x'^i/^; a—M, a?, 63, a^b^, 

— a2 jf and — a b^. 

12. x^—i/, x^, x^y^, —x^yi, | ?/f , J 2:t, — |- ?y^, and 

.21 .2_1_ 

rr^ 7/4. 

13. J - .2 6f :2:2 + ;3 7; -^i _ ^1^ J ^1^ _ il ^2^ and J b^ xi 

14. 1^--^'" -y «--6-"' + |6, 'Sa-"\ -rjb-'^, and 

15. «"+« + f/"6'« + (fc"»6'* + 6'"+", a.'", ft"*, rr", 6"", and 

24. Example 1. Multiply m-{-ti by a;+y; also m^n hyx — y. 
(m + w) (v+y) means that a: + ?/ is to be taken m + n times. Ifence, 

Process 

(m + n) X 0- -T-y) = (^' + y) + (^' + 2/) + C-^' + y) 4- • • . • taken w + n times 

= [(*' + ?/) + (•^■ + ?/) + (x + y) + ... taken m times] 

+ [ (*■ + ?/) + C^' + y) + (^ + y) + taken n times] 

= (x -{- y) m + (.r + ?/) n 

= (1) Art. 23, ??* a- 4 m y + n x -\- n y. (1) 

Also, 
(m + n) (x - y) = (x-y) -{■ (r~y) + (x-y)-\- .... taken w + n times 
-■ [(-^ — ?/) + (t ~ ?/) 4- (?' — ?/) 4- — taken m times] 
4- [(^ " y) 4- (j" — ?/) 4- (r — 1/) -f — taken n times] 
= (x ~ y) m-{- (x - y) n 
— (2) Art. 23, nix — my -\- 71 x — n y. (2) 

Similarly, (7n -\- n + p) (x-\- y — z) = m x + 7n y — mz + nx + n y 

— 71 z -\- p X -j- p y — J) z 



ALGEBRAIC MULTIPLICATION. 43 

These results are obtained by multiplying each term ot the multi- 
plicand separately by each term of the multiplier, and connecting the 
products with their proper signs. 

Example 2. Multiply j*-i*+2x^-x-5 by x*-h3x*-\- 5. 

Process. 

x«- r«-j-2x«-a:-5 
X* + 3 x« + 5 



x" - x« -H 2 x« - x« - 5 X* 

+ 3z» +6x*-3ir*~3x8- i5x« 

+ 5 x« - T) j;5 + 10 x*-* - 5 x - 25 

xW -H 2 x» -I- 7 x« - « X* - 3 x8 - 16 x» + 10 x2 - 6 X - 25 

Explanation. Multiplying each term of the multiplicand by 
each term of the multiplier and connecting these results with their 
proper signs, we have x**' — r* -f- 2 x* — x* — 5 x* + 3 x* — 3 x* 4- 6 x* 

- 3 X* - 15 x« + 5 x« - 5 x« + 10 x2 - 5 X - 25. Umling like terms, 
for a simplified product, we have x^<> -f- 2 x* — 3 x^ -f 7 x* — 8 x* — 15 x* 
+ 10 x« - 5 X - 25. 

The process used in practice is shown above. The first line under 
the multiplier contains the product of the multiplicand and x*. The 
second contains the product of the multiplicand and 3 x*. Etc. To 
facilitate adding, write the several products so that like terms shall 
stand in the same column. 

Hote. It i^ convenient to arrange the terms of the multiplicand and multi- 
plier according to powers of some common letter, ascending or descending. 

Example 3 Multiply f a x + f x^ + ^ a* by f fl^ + § x^ - f a x. 

Solution. Arrange the expressions according to the descending 
powers of r. Taking the multiplican<l | x* times, we have x* + a x' 
-f ^ rt^x^. Taknig it — | n r times, and writing the proiluct so that 
like terms nhnll stand in the same column, we have — ax* — a*x* 

- 1^a*T, Again, Uiking it | a^ times, and writing the pnnluct as be- 
fore, we have ^a^ x^ -h ij^a* r + \ a*. Adding the partial products, 
we have x* -I- ^ a*, or arranging alphal>etically, { a* -f- x* 



44 ELEMENTS OF ALGEBRA. 

Process, 

f x2 - f a a: + f a^ 



— ax^ — a'^ x"^ ~ ^ a^ X 

+ ia^x^ + la^x + ia* 

Example 4. Multiply - Sx^' + ^y^ - .Sx^^^^ y'' + ^ + 3.3j:"'^ + ^ 
by — .2 j:™«/«-2 + 4 a;'"-' i/«-^ 

Process. 

3.3 a;*"*/" + 2 — .3 x"" + ^ y'' +^ —3 x"' ^'^y'' 

4 x™ - 1 ^'» - ^ — .2 a;"» ,y" - ^ 
13.2r*"»-i/«+i -1.2 a:-'"**/-'" - 12.00 x^^ + 'j/an-i 

- .66a;2'«?/2»,_^ 06a;2"» + i j,2«-i^ g^zm+a^^n-g 

13.2 j2m-1^2n + l_l 86^:2 '"y^»- 11.94x2"' + ! ^2«-I_^ (5 a,2m + 2^2«-2 

Explanation. Arrange according to the ascending powers of x, as 
shown. The product of the multiplicand by 4x'"-' ^" ~ ^ gives the 
first partial product, as shown on the first line under the multiplier. 
The product of the multiplicand by — .2x'"^"-2 gives the second 
partial product. Taking the sum of the partial products, we have 
the product required. Hence, in general, 

To find the Product of two Polynomials. Multiply the 
multiplicand hy each term of the multijplier, and add the 
partial products. 

Exercise 16. 

Arrange the terms according to the powers of some 
common letter, and multiply : 

1. r/2 J^h'^-ah by « 6 + ^2 + ^2 . a'^-2ax ^ 4.x^ 
by «2 + 4 .^2 + 2 a a^. 

2. x^ -\- y^ — x^ y"^ by x'^-\-y'^\ x + y -{- x — y by x-^-y 
-x + y. 



ALGEBRAIC MULTIPLICATION. 45 

3. y— 3 + /y2 by 2/-9+2/2; a^tj — azi-ij^ — a^ by y-^ a. 

^ J a;2 - I X- - f by J a;2 ^ I ^ __ 1 . 1 6 «2 4. 1 2 a 6 
+ 9&2 by Aa-.:n, 

5. x^— i/-\- X — fj by x^-^ y^-\- x — y \ ^s^ — ax — ^a^ 
by \ x^ — ^ax -\- ^a^, 

+ \d^ by 2 a;^ + « 2: — ;| ^3. 

7. n7^-Dx'^-x^^-2^-x + 2hyx^-2x-2, 

8. 3a2-2a3_2a + l + a* by 3ft2+ 2^3+ 2a + l + ^*. 

9. 1.5 2:8 + 1.5 2^2 + .5 a:* + .5 2: + 2:^ + 1 by a^ - .5 a; 
+ 1 + a:* - .5 2:8 

10. 1 + 9 a + 5 a3 + 3 a* + 7 a2 4- a^ by 4 a2 - 3 a8 
4- a* 4- 4 - 4 a. 

11. 4 2:2^24. 82:^3+16/ + 2a:3y _^^ by 2: - 2 y. 

12. 2^12- a:3^6 4. 2^6^- 2:87/24. y8 by y«-|- 2:8. 242^2 

— 3^ y — X i^ -\- x^ -\- y^ by x -\- y. 

13. rt2 _^ ^2 _^ ^ _ ^ 5 _ rt c _ ^ c by rt + & + c. 

14. ^2 _^ ^,2 _^ ,.2 4. J c + a r - a 6 by a + fe - c. 

15. a6 + crf4-ac4-6c^ by ab-hcd^ac — bd. 

16. i2 4. y2 _ 3 3^ _ y2 by 2 a: + 2 3^ - 2 (2: - y). 

17. 3 (m + n) — .1 X (a + h) by a - b -^ .1 (m ^ n). 

18. a2f^-\-bx''-\-r by a2:* + 62;"+r; 2;^ + yi by 2:^ — y^- 

19. a- + 6" by a'" + b"; oT 4- 6* by a" - 6*; 2:2 _,_ j by 
a:i + bl 



46 ELEMENTS OF ALGEBRA. 

20. Sx"^-^ - 2/-' by 2x- S'f; ax''' + &^"+ ahx 
by a x^ — bx^ — 1. 

21. 'Sa^^'x+'Sa^y + a''' by a'"- a" + 2:; x^-y-i by 

2^2 — y. 

22. .2ai-.3&t hy.2ai+.3bh^xi + xUji-hy^ hy x^-yk 

23. a;^ ?/~t + y~^ + x^y"^ + a;^ by x^ — y~i. 
Find the product of: 

24. 1 + 2;, 1 + 2;^ and 1 + x^ — x — a^. 

25. a; — 2 a, X — a, x + a, and a: + 2 a. 

26. 3 2; + 2, 2 a: - 3, 5 :z: - 4, and 4 a; - 5. 

27. ^2 — :r + 1, a;'-^ + :r 4- 1, and x'^ — x'^ -{- 1. 

28. rr^ — a x-{- a?, x^ + a x ■}- a?', and x^ — a'^ a;^ + a*. 

29. « + 6, a - 6, 3 a + &, and a^ -2o?h - a}?' \ b^ 

30. rr + &^ «"*-&", ft''"+a"*6"+?)'", and a2'«_,^«^H + ^a«^ 

25. A Binomial is a compound expression of two terms ; 

SiS, a — b; ab + 2b\ 

In each of the following products, observe that : 

2a; + 3 2 a: + 3 

2x + 5 2a; - 5 



4a;2+ 6 a; -~ 4x2+ ^^^ 

10 a; +15 -10 a; -15 



4x2 +16 a; +15 4^2- 4 a; -15 

2a;-3 2 a; -3 

2x + 5 2a; — 5 



4a;2- 6 a; 4^2- 6 a; 

10 a; -15 -lOx+15 



4x2+ 4 a; -15 4x2-16x+15 



ALGEBRAIC MULTIPLICATION. 47 

I. Thejirst term is the common algebmic term of the binomials 
multiplied by itself, or the square of the common algebraic term. 

II. The second term is the al«;ebraic sum of the other two terms of 
the binomial expretssious multiplied by the common algebraic term. 

III. The last term is the algebraic product of the terms which are 
not common to the binomial expressions. Hence, 

To find the Product of two Binomials, having one Common 
Algebraic Term yidd toy ether the nfpiare of the common 
tertn, the abjebraic aum of the other two tervis multiplied by 
the cmiimaii terniy aiul the algebraic proditct of the terms 
which are iwt common. 

In general, {x -\- a) (x ±b) — 2^ + {a ±b) x ± ab (1) 

(x-a){x±b) = x^-\-{-a±b)x^^ab (2) 

In which a, b, and x represent any numbers. 

Hotel: 1. It is of the utmost importance that tlie student sliould learn to 
write tlie products of binomial expressions rapi<lly, by inspection. 

2. To square a monomial, multiply the numerical coefficient by itself ^ and 
multiply the ejptment of eacli letter by two. The proof is evident. Thus, the 
square of 2 a* 6» =- 2 X2 a* ^ * A* >< « - 4 ofta-. 

Also, (36-"ar«)a = 3 X 36-»x2a*« ^a - 96-2««a;2m. 

Examples. Write the product of the following by inspection : 
(2 a; + 7 y) (2 I - 5 i^); (a - 9 6) (a - 8 6) ; (a - 6) (a + 1 3) . 

Solution. Squaring the common term, we have 4x^. Taking 
the algebraic sum of the other two tenns, + 7 y and — 5 y, we have 
+ 2y. Multiplying this sum by 2 a;, we have + 4a:y. Taking the 
algebraic product of the terms not common, + 7 y and — 5 y, we have 
— 35 y*. We thus obtain 4x^ -i- 4 xy— 3b y^ for the product. 

Similarly, (a-96) (aSb) = a«+ (-96-86) X rt + (-96) X (-86) 

= a3-17a6-|-72 6^. 
Also, (a-6)(a+13) = a2+(-6 + 13) X a-f (-6) X (4-13) 

= a2 + 7a-78. 



48 ELEMENTS OF ALGEBRA. 

Exercise 17. 

Write, by inspection, the products of the following : 

1. (a -3) (a + 5); (6+6)(&-5); {x + 4) {x + S) 
{x - 4) (0^ + 1) ; (^ - 7) (x + 2). 

2. (x - 8) (^ - 6) ; (a + 9) (a - 5); {a- 8) (a + 4) 
(2x-4:) (2 a^ - 5) ; (3 ^' + 7) (3 ^ - 5). 

3. (0^3-37/2) (^3_ 4 2^2). {x-7y)(x + Sy);{a"^-l){a-+2) 

(3a:5-5)(3a:^-4). 

4. (2 a2 2/3 + 4) (2 ^2 f _ 8) ; (3 a a; - 4) (3 a 2: + 7) 
(a:3 + 3 a) (a;3 - 4a;) ; (ai^ - 3 a2) (a;^ + 2 a^). 

5. (2a; + a)(2a;-2a); (2:c"+ 5a)(22;"-3«); {Sx-2y) 
(S X + y) ; {- 6 m + 2 2^) {4.m + 2 x^). 

6. (:r-a)(2:-5a); {a-5b)(a + Sb); {a^-2x){a^-6x); 
(5xio+3a2)(5:z;io_4a2). 

7. (32/2-5a:^)(2 2/2-5:?;5); (3 a^ + 2ab)(3a^-4:a¥); 
(a" + 3) (a» - b). 

8. (4 a + 6) (4 a - c) ; (2 & - 5 a) (2 c - 5 a) ; (a y 4- i^;) 
(a.^ + i^); (af-l)(al+|). 

9. (2 2:^ + 1) (2 xi 4- 12) ; (2 a^ - 3 ax) (2 a^ + b) ; 
(x- .Sx^y''){y- .Sx^y"*). 

26. (:r4-7/)(2;- y) = x'^ + {y - y) X x -]- (i- y) X (-y) 
= 2^2 — ?/2_ Xn which a; and y represent any two numbers. 
Hence, in general, 

To find the Product of the Sum and Difference of two 
Numbers. Take the difference of their squares. 



ALGEBRAIC MULTIPLICATION. 49 

Examples. Find the product of (2 a"* -f- 3 b-") (2 a"» - 3 &-*) ; 
(8;)* 4- ll2*)(8;>*- Hz*). 

Solution. (2a« + 36-") X (2a" - 3 ft-*) is the square of 
2 a*", or 4 a* •", minus the square of 3 6"", or 9 6-**. Therefore, 
(2 a"» + 3 6-*) (2 a" - 3 6-*) = 4 o* »» - 9 6- «*. 

Similarly, (8/)* 4-112*) (8p* - 11 z*) = 64/)« - 121 z. 



Exercise 18. 

Write by inspection the product of the following : 

1. (2x-^'Sy)(2x^Sy); (x -{- 2ij){x-2y); (5 + 3 a;) 
(5-3a:); (5a:+ 11) (5 2; -11). 

2. {2x-h l)(2a;-l); (2x+ 5){2x-b); (5xy + 3) 
(5 a; y — 3) ; (c -f a) (c — a). 

3. (c2 + a2) (c2 - a2) ; (m n -H 1) (wi n - 1) ; (« y^ 4. j) 
{af --}))■ (a2r2+ l)(a2a:2_ 1) 

4 (a:* + 7/) (ar* - /); (1 - pq) (1 -f- pq)- {m - n) 
(m^-n)\ (a"* -fa") (a* -a"). 

5. {bxr^^^y^{oxy-^-V4.f); {h^+^f)(^2?-?>f)) 
{2!^-Zx){j^+ 3a;). 

G. (2aa:4- fey)(2aa;-6y); (m"' + 7i-») (m"'- 7^-»); 
(10 a-" - 13 6—) (10 a— + 13 6—). 

7. (mi + rA) (mi - ni) ; (4 ai- 20a;io) (4ai + 20 a:iO) ; 
(ai-6-f)(«i + 6-i). 

8. (11 a:i + 30 ?/*) (11 a:* - 30 y*) ; (15 a2 6^ - 16 a* 6^) 

9. (i«6-2+56-Ja:-i)(Ja6-2-i6-ia;-i); (a+6)(a-6) 



50 ELEMENTS OF ALGEBRA. 

10. (ah+l)(ab~l)(aH^-\-l); (2a'"+ 4a")(2 a"'-4a") 
(4^2"^+ 16^2"). 

11. (5 a^ + 6&2) (5 a^ - 6h^) (25d^ + 366*) ; {a-^ + aH^) 

12. (rc-l + x-Uj) (x-^ - x-'y} (^x-^ + x'' f} ; 

(f cr- + i If) (f c- - |-6«) (f f c-'^- + }f 62"). 

Queries. In finding the product of monomials, why add expo- 
nents of like factors ? What is the product of a^ and a^ ? Prove it. 
Why is the product of an even number of negative factors positive ? 
How prove (1) and (2) Art. 25 % 

Miscellaneous Exercise 19. 

Multiply : 

1. 2 ^2" - a" + 3 by 2 a2- + a" - 3 ; 5 + 2 a;2« + 3^ 
by4^«-3^2a 

2. ft^ + 2 a^" - 3 by 5 - J a" + 2 ^2* ; J a;i - 5 + 8 a;t 
by \x^ + lx~'^. 

3. 3 ^1 _ a - a^ by f a^ + a"! - 6 a-i ; 2 a*""^ &-"' 
+ a-'^ 6^ by 3 a'" ?^'^ - ««^ Ir^K 

4. ^x''if—'ix-''y-^ by 4 ^«?/ + 5^2a^26. ^t" + «-t" 
by ai" + «~^". 

5. .3 ft* - .02 ft36 + 1.3 «2^2 + .5 a2,3_ 1 2 h^ by .3^2 
- .5 a & - .6 2>2. 

6. 1 - 2 ^^ - 2 ^i by 1 — .T6 ; al - 8 «-t + 4 a-^ - a^ 
by ia~^ -\- a -\- I a-\ 

7. 2x^-x^-3x-^ by 2 x'^ - 3 x~^ - x-^ ; a" - 1 
+ ft- " by a^ + ft~i 



ALGEBRAIC MULTll'LICATlON. 51 

8. ^■""■'■^ - x-" + ' - X-+ x"-' by 0;" + '^ - jJ" - ^; + 1 ; 
a,'»+3a;"-*-2a,"-^ by 2 .x' + ' + it:- + » - 3 x*. 

lU. 3^- 2a;'" + ' - 5a;'" + =»+af* + ^ by 3a;"-3 + 2 3;"-* 



11. a:" + ^- 3x-+*+ x*+''- 2ic- + * by 2a;'-"+ Sic^' — 

12. 5a;"-V-*-'-2a;--^/-^'-u;**-'/+' by 3 a;*+ */"' 

13. //«.' + ' — 3m'*?i+m'-^7i2—?/i'-^t^ by m*"-^— 3m-?i 

14. 2.«;"+V"'4-3a=*'"'/'"^-a;"+Y'"' + 4x«+V'"' 
by 2./;■+y-'**'-3^-•^Y-'' + .tV"'' + 4x"-y-^ 

15. x-+*y-* + x'-^*if-' - 2a;- + V~" - 4a:- + 'y-* 
+ 4a;»— y^"- 

16. (y+ a;-)(y- .»:-'") ; (i^^r -f •^"VXi^rHf^c-/^). 

17. (x'' + y~)ra:'*-y-); (a:i - 5) (xi ^ 4), (7 x ^ 3y-^) 
(7x+3rO. 

18. (4 X' - 5 x-^) (4 a; + 3 x-^) ; (f A b'^ - ^^ a^I)^) 
(|cU-t + ^a^65); (a- 4- 7 + 3a-'')(a~ + 7-3a-''). 



52 ELEMENTS OF ALGEBRA. 



CHAPTEE V. 
INVOLUTION. 

27. Involution is the operation of raising an expression 
to any required power. 

hivokition may always be eflected by taking the expres- 
sion, as a factor, a number of times equal to the exponent 
of the required power. 

It is evident from the law of signs that even powers of 
any number are positive ; and Ihat odd powers of a number 
have the same sign as the number itself. Thus, 

(— m* uY = (— m* u) X (— m^ n) 
(- m* 7i8)8 = (- m*?i8) X (- w* n^) X (- m* n^) 

= -m< +4 + 4^8 + 8-1-8 =:-ml2u». 

(- 3 m8 ny = (- 3 m^ n) X (- 3 m^ v) X (- 3 w^n) X (- 3 m^ w) 

^ + 31 + 1 + 1 + 1^8 + 3 + 3+3^1 + 1 + 1+1 = 4. 81 mi'-^n*. 

(a** 6<^)~ = a*^ b" X oT^ ¥ X a"^ b*" X ton factors 

= (a*" X a"* X a"* X .... to n factors) X (6^^ X 6*^ X 6*^ X ... . 
to n factors) 

/'Qm+m + m+.. .. ton terni8\ y^ /'^c + c + c+.... ton termsA 

— qM X n ^ ^c x n 

— Qmn ^cn^ where c, m, and » are positive integers f a and 6 
may be integral or fractional, positive or negative. 

Similarly, (a"*6<^# joO" = o*«"6<^"cZ*« .... ;?♦•". Hence, in 

general. 



INVOLUTION. 63 

To Baise a Monomial to any Power. Multiply the exponent 
of each factor by tlie exponent of the required power y and take 
the product of the resulting factors. Give to every even 
poiver the positive siyyi, ami to every odd power the sign of 
the monomial itsdf. 

Notes : 1. Since, aw - - 1 X a"», the nth power of - a™ -» (- 1 X a"*)" 
— (— 1)* X «"•». Or we may write ± a'"»», for the nth power of - a»»», where 
the positive or negative sign is to be prefixed, depeudiug upon the value of n 
whether an even or odd integer, in being positive and integral. 

2. Any power of a fraction is found by taking the required power of each of 

Uliistrations. 
(-3x'»/)« = -3''<«a:«'<V'' =-27x«y« 

Exercise 20. 
Write the results of the following : 

1. (4aH*)2; (3a668)3; (2 a; V)^ (^a^V^c^f; (.1 a'»6-)^ 

2. {1 a^V^f) (llaJ2c8(fiO)2. (-3cx3/2«)8; {ZaH'^iff; 
{5abcc^y^z^y. 

3. (-2a2)c2^7/*)8; (-abcdxy^- (-a^l^cf; (-^^O*; 
{SaPc^; {-2aV^f. 

4. (lXa<68c2^)n; (-2a2"6"»)^ {-Zxyzf\ (x^^^^^ "-)•"; 

5. (-2a''66«.)8. (^•)«. (a«)a. (ft ft -ic- 2)6; (m"7i— )•"•; 

(-2)8; (-«)«"; (-ir. 



54 ELEMENTS OF ALGEBRA. 

6. n(2 a b'^ c n "i)*; n {n^ m^*"")"; m (m" a"^)"; 

7. 2 a(- 3m7i3a;*)3; m(- 3 aiojs^^e^*)*; a>^ (3 a-^J-^; 
a(a«-')«. 

8. C2x^y\zhy\ (-ra;V"^T; (-3ain-"c)"; 
(_ 3«-''6't^iyA)6. 

Affect the following with the exponent 7 ; that is, raise 
each to the 7th power. 

10. {-x''y'^f-{ahh^f;{^^a})xy\ [{-x'^yf ', {-T^mTf. 
Write the nth. powers of: 

11. mia-'^ciyix-yy; (a-3 d)^ '' (x-yf ; 3 (a-b-\-c-\-d) 

{a — xy. 

12. a&c(a-6T(^ + 2/ + ^T; a''{x-y-^zf''\x-y'^f''\ 

28. It may be shown by actual multiplication that : 
{a + hy =a-H6H2a&; 

(a-hy =a^+h^-2ab; 

(a + b-\-cy =a2-}-62_,_c2+2a6 + 2ac + 26c; 
(a-b-cy =a^+b-^-\-c^-2ab-2ac + 2bc; 

{a-\-b + c-\-dy=a^+b^+c^-\-d^+2ab-]-2ac + 2ad + 2bc+2bd-\-2cd; 
etc. etc. etc. 

In each of the above products, observe that the square consists of : 

I. The sum of the squares of the several terms of the given 
expression. 

II. Tvnce the algebraic product of the several terms taken 
two and two. 



INVOLUTION, 55 

These laws hold good for the square of all expressions, 
whatever be the number of terms. Hence, in general. 

To Square any Polynomial Add together the squares of the 
several tertns aiid twice the algebraic prodtict of every tvH) 
terms. 

Example 1. Square 3 a» — 4 x*. 

Solution. The squares of the terms are 9 a' and 16 x^®. Twice 
the algebraic product of the terms is — 24 a^ x*. 

Therefore, (3 a» - 4 x»)« = 9 a« + 16 x" - 24 a«x». 
Example 2. Square 2 x' - 3 x* - 1. 

Solution. The squares of the terms are 4 x*, 9 x*, and 1. Twice 
the algebraic product of the first term and each of the other two 
terms gives the products — 12 x* and — 4 x*. Twice the product of 
the second and third terms is 6 x^. 

Therefore, (2x»-3x2- l)a = 4x»+ 9x* -I- 1 - 12x»-4x« + 6x«. 

niuatrations. 
(2 a"» - 3 x-»)« = (2a "•)2 -|- (- 3 x— )«+ 2 (2 a"») X (- 3 x^") 
= 4 a*" 4- 9 r-«" - 12 a"» x"*. 

(x-V-Ky-*+|y'-iy)'=(^V)"+(-iar-tr*)*+(|y^+(-iyy 

+ 2(x-V) X (- ix"y-«) + 2(r-«y'') 
X(§y'H-2(x-V)X(-iy) + 2(-ix"y-«) 
X(|y«) + 2(-ix-ir*)X(-iy)+2(fy») 
x(-Jy) 

+ |x-«y*+t-|r-«j/*+»-§x"y + ix*y-'» 



56 ELEMENTS OF ALGEBRA. 

Exercise 21. 

Square, by inspection, the following : 

1. x + 2; m + 5; n-\-7; a — 10; 2x -\- Sy; a + Sb; 
a — Sb; 2x — Sy. 

2. X + 5y; 3x — 5y; 2a + ab; 5x — Sxy; 5abc — c; 
xy-^-2y^; a™ + 3 6"". 

S.2x + Sa^; xy + x'^; 3 a-2 -f 5 a^-^; 1-x; 
1 — cy; m — 1; ab"^ — I ; -J a"— .05. 

4. 1 a 6-2 + |6-ia;-i; fff-r^'^ | a-« _ 2 j-»«; 
ic-f 3/-f + J; .0002a;'" + .005/. 

5. I m^ n^p^ — ^mrf; xy + yz-hxz; 2x^ ■]- Sx — 1; 
x^-2x+l; x^ + 2x-4. 

6. 2a^—x + S; a^—5x—2; x^—2xy + y^; 4:n^+m^n—7i^; 
x^ - 3x 4- 2. 

7. xy — 2n + 1; m — n — p — q; a^ — 2a^ -\- 2x — 3; 

1 + X -}- x'^ + x^; x+Sy+2a — b. 

8. 2a^-Sa^-x + 3; x - 2y - Zz + 2n\ wT ^- tT 

9. ^a-2b-V\c',xf-y- + \a-\b;la^-x + l', 

10. l^lx-\x',la^-\x-\;\ar-\a-+\xy', 

2 a;t + 5 a:i + 7. 

11. '^x^-2x^ + \x^-x-^; m^"-f a;i«2/-t*-|^'^-3; 
2i-3i. 

29. Any Power of a Binomial. It may be shown by 
actual multiplication that: 



INVOLUTION. 57 

(a + 6)3 = a3 + 3 an + 3 a?;^ + 68. 

(a - 6)3 = a8 - 3 a26 + 3 a 6^ - JS; 

(a + 6)* = a* + 4a36 4- 6 a^l^ + 4:ah^ -^ 6*; 

(a - 6)* = a* - 4 a36 + 6 a2 62 - 4 a68 + 6*; 

(,f + 6)'^ = a^+ 5a*6+ 10 a8 62 + 10 a263 4- 5a6*+6«; 

(,i -6)'^ = a6-5a*6+ 10a362- 10a263+ 5a6*-6fi; 
and 80 on. 

In each of the above products we obsei-ve tlie following 
laws: 

I. The number of terms is one more than the exponent of 
the binomial. 

II. If both terms of the bin^omial are positive, all the terms 
are positive, 

III. If the second term of the binxrmial is negative, the 
odd tei-ms, in the product, are positive, and the even terms 
negative. 

IV. TTie first and the last terms of the product are respec- 
tively the first aiul the last terms of the binoinial raised to 
the power to which tlie binomial is to be raised. 

V. The exponent of tlie first tei^m of the binomial, in the 
second term of the product, is one less than the exponent of 
the binomial, and in each succeeding term it decreases by one. 

The exponent of the second term of tlie binomial, in the 
second term, of the product, is OTie, and in each succeeding 
term it increases by one. 

Thus, omitting coefficients, 

(a + 6)« = a« + a^b + a*62 + aS^s ^ «2 j4 j^ ab^ ^ 6« 

VI. The coefficient of the first and the last term is one, 
that of the second term is the exponent of the binomial. 



58 ELEMENTS OF ALGEBRA. 

The coefficient of any term, multiplied by the exponent of 
the first term of the binomial in that term, and divided by 
the number of the term, will be the coefficient of the next term. 

Notes : 1. The sum of the exponents in any term of the expansion is the 
same, and is equal to the exponent of the binomial . 

2. The coeflScients of terms equally distant from the first terra and the last 
term of the expansion are equal. Thus, we may write out the coefficients of 
the last half of the expansion from the first half. 

If one or both of the terms of the binomial have more 
than one literal factor, or a coefficient or exponent other 
than 1, or if either of them is numerical, enclose it in 
parentheses before applying the principles. Thus, 

Example 1. Expand (2x'^-5a^xy 

Process. 

(2 a:8 - 5 a2 a;)* = [ (2 a;3) - (5 a2 x) ]4 

= {2x^y~4{2x^f{5a^x)-\-6{2xy{5a^x)^-4{2x^){5a^xy 

+ (5a^xy 
= 24a;i2_4 X 2^x^ X 5a^x-\-6X 2^x^ X b^a^x^- 4 X 2a:8 
X5^a^x^ + 5^a^x^ 

= 16a;i2_4X8a:«X 5a^x-{-6 X 4x«X 25a*a;2-4x 2x8 

X 125a^x^+626a^x^ 
=z 16a;i2_160a2a;io + 600a4a:8_1000a6x«+625ft8a:* 

Explanation. In the expansion the odd terms will be positive, 
and the even terms negative. The first term is (2 x^y, and the fifth 
or last is (5 a^xy. The exponent of (2 x^). is 4, and in each succeed- 
ing term it decreases by 1. The exponent of (5 a^x) is 1, and in each 
succeeding term it increases by 1. The coefficient of the second term 
is 4. For the second term we take the product of 4, (2 x^)^, and 
(5 a^x). To find the coefficient of the third term, we multiply the 
coefficient of the second term 4 by 3 (the exponent of (2 x^) in that 
term), and divide the product by 2 (the number of the term), and 
have 6. Hence, the third term is 6 (2 x^y (5 a^x)''^. The coefficient 
of the fourth term is found by multiplying 6 (the coefficient of the 
third term) by 2 (the exponent of (2 x^) in the third term), and 



INVOLUTION. 59 

dividing the product by 3 (the number of the term). Hence, the 
fourth term is 4 (2 x*) (5 a* a:)*. Performing operations indicated, we 
have the required result. 

Example 2. Raise 1 — § x" to the fifth power. 

Process. 

(l-§x")» = (l)»-5(l)*(|x«) + 10(l)»(§x»)«-l0(l)«(fz»)» + 5(l)(fxn)* 

= 1»-5X l*x|a*+10X l«x|x2--10X l«X^x»« + 5X 1 



Exercise 22. 
Expand and simplify the following expressions : 

1. (a - 6)7; {a 4- x)^; {a^ ~ ac)*; (a^ - 4)^; (2 + a)*; 
(«-l)^ (1 -aY; (2a-Sby. 

2. (xh - 3)* ; (ax-S x^f ; {x - 3)^ (2 a^z + 3 62^)8; 
(2ax-\- Sbyy. 

3. (« + 2)e; (a-2)«; (2-Ja)*; Cja-36)*; (Ja + }6)*; 
(a + 6)W. 

4. (ai - 2 - a-^y ; [(2: + y)H {x - yff\ (1 + a + a^)^ 
- (1 - a + 2 a2)'f 

5. (a + 2i)*-(a-26)*; (3 - 2a + a2)2 - (2 - a)*; 
(3i + 5i)2 _ (2i - 3i)2 

Queries. How prove (— m)* = ± m", according to the value of 
n, whether an even or odd integer ? How prove the method for 
squaring any polynomial? How prove the laws for raising a bino- 
mial to any power? 



60 ELEMENTS OF ALGEBRA. 

CHAPTER VI. 

ALGEBRAIC DIVISION. 

30. Division is the inverse of multiplication, and is the 
operation of finding the other factor, when a product and 
one of its factors are given. The product is now called the 
Dividend, the given factor is the Divisor, and the required 
factor is the Quotient. Thus, 

since a^ X a^ = a^ .*. a^ ^a^ = a^-^ 

since a-^Xa^^a^, .*. a^-^a^ = a-^\ 

since a^ X a-* = a, .-. a-^a-* — a^; 

since a"*-" X a" = a"*, .*. a"» -r- a" = a"»-"; 

since «"»+" X a-" = a"*, .*. a™ -^ a-" = »*"+" ; 

since Sa^ft* x 2a-26 = 6a6^ .-. Qah^ ^^a-H = ^a^b^; 

since 9a-362 x 3a*65 = 27a6^ ... ^1 aU' -^Za^h^ = ^a-H^; 

since 5 a*6"~^ X 4a~*6^ = 20aH*, .•. 20a^h^^Aa~^h^ = baH~^ ; 
etc. Hence, in general, 

To Divide a Monomial by a Monomial. To the quotient of 
the numerical coefficients annex the literal factors, each taken 
'with an exponent obtained hy subtracting its exponent in the 
divisor from its exponent in the dividend. 

Illustrations. 

a^h^c^m^-^ a^b^c^m^ ap-^b'^-^c^-'^m^-'^ =abc*m. 
63a-26V5^ 7«-36c'» = 9a-2+3i2-ic5-4 =()abc. 

1^ 2A2 . «^ 5a2i2-i ^a^b ,. ^ „ 

I5a^b^-^6bc = — = (Art. 2). 

2c 2 c ^ ^ 



ALGEBRAIC DIVISION. 61 

Exercise 23. 



Divide : 



1. 3a362 by ab] I6a*i^ by 3aH^; 20a^}^c^ by 5a6V; 
Smi by 5 m^. 

2. tT^ by ?i~i^; a* by a*"^; a^j-s^n y^y ^35-2^2. ^m+i. 
by a"* — ; 2'+' by 2'-'. 

3. 15 a-t 6-i a^ by 9 a-2 6"! ic^. ^ ^i fti by f (A ji ; 
21a*m2ic' by Tama:*. 

4. 24a"j9'" by 3a>"; 36a'"mV^ by 9amyri*; «*'+y-* 
by a^/. 

5. (x - y)6 by (a; - y^; (a - c)*+8 by {a - c)*-i ; 
|fe*^^•- by f 6/H^. 

6. (6a3 62,; X iSftSJV) by (S^a^'c^ x 2a*c8); a*"' by 
a*"; (2 77i7i")2* by (2mn«j^ 

31. Only a positive number, + a, when multiplied by + 6, can 
give the positive product +a6. Therefore, +ab divided by +6 gives 
the quotient + a. 

Thus, since aX6 = + a6, .•. 4-a6-f + 6 = -f-a; 

since aX-6 = — a6, .*. — a6^ — 6 = -|-a; , 

since — aX6 = — a6, .*. — a6-^ + 6 = — a; 
since — aX— 6 = 4-a6, .*. -{- ab-. — b = — a. 

Hence, in general, 

Law of Sig^. If the dividend and the divisor have the 
same siguy the quotient is positive. If they have opposite 
signSy the quotient is negative. 



62 ELEMENTS OF ALGEBRA. 

Example. Divide 12a"» by — 4 a". 

Solution. Since there is a factor 4 in the divisor, there must be 
a factor 3 in the quotient, in order to give a product of 12 in the divi- 
dend. Since there are m factors of a in the dividend, and n in the 
divisor, there must be m — n factors of a in the quotient, in order to 
give a product of a"» in the dividend. Hence, 120'" -f - 4 a*^ = — 3 a"*-", 
because only a negative number, — Sa"*"", when multiplied by —4a" 
can give the positive product, 12 a*". 

Illustrations. 

- ISa^mHS-f 3a2m*62 - - b a^-^b^-^m^-'^ = ~5a^bm^. 

- 5 x^^y^z^ i- -10 x^y^z^ = + ^ cc^o-s ^^8-5^6-3 ^ ^ ^x^y-^z\ 

ia'{a-by{x + y)^^ -4:a{a-b)^{x + yy= -|a"-^(a-6) (x-t-i/)"*-". 



Exercise 24. 

Divide : 

f. 6^ by 3^; -20aH^cJ by lOahc; 35a^^hy-7a^; 
-laHc^ by -7aHc^. 

2. 27ax* by -9^4. _|a6^,6c6 by ^aHc'^; .^aH^^c^^ 
by fa^^iici*; 12^2"?/ 2 by -f^j''?/. 

3. Z\7n?n^3iP- by -2i m-i7i-3^-2; - 5| m-^j^-iyio by 
*lj2ga2m3x-4?/; 3.2Jrt-s^?/S by 2.Qt2\a-^xy\ 

4. .O^a^wiV^* t)y -.0|a2^2/2^3. _9.3m3«^2-^«-32/§c 
by .3m3«+ix'*-4 2/K 

5. .^x'^ifhy-ix'y-'^'^- -J(a fe)3c8 by .6(a-&)2ci0; 
~ .3ai'"ft^ by -.2 a" 6". 



ALGEBRAIC DIVISION. 63 

6. - .375 xi ?/U-^^ - y^^ by - i^ oc^ y (xi - i/l)l . 
8m-Si-^r-02y7 by 9 m ^ ,. "- x-^ y- \ 

7. -1.2aiO(jc-y)"r» hy .^a^{x-yfz^'^\ m-^n^x-yY 
(y-zY by w 2«n2''(a;- 7/) '•(?/- 2)i^. 

Simplify the following, that is, perform the indicated 
operations : 

X -~.5a2''62«c-2'. 

9. {a-H*-^2ab) x -2a2fe-2 x (- .Gaifti -=- - .3aUi). 

10. (.3a-'"6-'*c-''H- .03a"'6-c'') -r 1 j «-3'"6-3»c-3''A:. 

11. (4§«-»6(ijc-2-f-lia-U-3rf*) 

X [6 a2c- irf3 ^ ^ (84 ^8 j8 c -f- 7 a* b^ c^)]. 

12. (ic""^'* X a-U-i.i-"-") x (rtUa^^-^^^i -!-6ir*'"7/-t). 

13. (-Ua^H^c-^^-7aH*c-^) 

-=- (28 a-^iV ^ - 4 a-'^b-^c-"). 

14. (1.7«-i6-^cijc2-M.l«-2i-ia:8)x (a"'i*c-6^al62c3). 



32. Since (a •^- b) m = a m + b m, .-. (am + bm) -r m = a + b. 
Since (a - b) m =■ am — bm, . • . (a m - 6 w) f m = a - ft. 

Since (iy-2y«2-3x»ir') X -3xy»= -3x«y< + 6xi/*2 + 9x*^, 
.-. (-3x»y*4-6xy»2 + 9x<y)T -3xi/» = 2r/-2ya2-3x»y-». 
Hence, in general, 

To Divide a Polynomial by a Monomial. Divide each term 
of the dividend by the divisor, ami add the results. 



64 ELEMENTS OF ALGEBRA. 

Exercise 25. 

Divide : 

1. 2a?+ ^a^y ~ 8a^y^ by 2a^- 21 m^n^ - 1 m^n^ 

— 14 myi + 63 by 7 mn. 

2. a^hc -- a? b^ c2 - a^ W' c^ + a^ b^ c^ by a^bc; 42 a^ 

— 1.1 a;2 + 28 ^ by .7 x, 

3. 28a3 + 9a2-21a + 35 by 7a; 4: a^ b^ - 16 aH^ 
+ 4:aH^ by - 4 aH^. 

4. 66t262c3 _ 48^264^2 + 36^2^2^! _ 20abc^ by 4a&ca 

5. 2.4 m27i2 — .8 mht^ — 2.4 m ?i2 + 4:m^n^ by .8 m n ; 
icf — x^ y^ by ici. 

6. -"^a^^^ab-^ac by -1.5a; .5 m57i2_ 3^3^* ^y 

— 1.5 m^v?. 

7. - 72 o^ c2 - 48 a} ^10 + 32 ^2 c^ by 16 a^ c^; 3.6 n* 

— 4.8 rtf by 4 n\ . 

8. 11^2^3^ 3 a; 71 -2.4 2/2 by .'^xy\ .09 7^1*- 2.4 m^ 71 
+ 4.8 «i5 by .03 wi* 

9. - ft"* + 2a-"' - 3a" by - a^\ m"+i- 7/i"+2 + 77i'*+3 

— m"+* by m?. 

10. 2.1 a 2:2 y"* 4- I.4a3;:c4^"- 2.8a5^2^P by - ,1 axy"". 

11. a'"53_^-+ij2_|_^n-2j by -ab; -2a^a^-j-S.5a^x* 
by 2.3^ a^x, 



ALGEBRAIC DIVISION. 65 

12. 2.25 a^x - .0625 aire - .375 a ex by .'67^ ax; 
llxi-33a;* by 11 x*. 

13. 72 wt - 60 mi ni + 12mi ni - 6mT^ni by 24 mi. 

14. 36 (x - yf - 27 (x-y)3 + lS{x-y) by 9(a:-y). 

15. -123ry*2'S0sf''-^^y^z+10SaP^y2f-^^ by -6^y*^ 

16. m" (x — y)« — 7;ia(a; — t/)" by m* (a; — y)*. 

17. {x-hyY(x-yy-\-(x+yy{x-yy by (x-^yy{x-y)\ 

18. -2.5m2+1.6m7i+3.3m by -.83m; a-^i-ai^ift^+a ^ 
by a-w. 

33. It may be shown by actual multiplication that : 

(TO+n+/>) (x+y-f-z) = ma:+TOy+m2 + nx + nT/ + nz+;)a;+/>y+/)2. 
.♦. (nu:^-my+'nu-\-nx-\rny-]rnz-\-px+py-\-pz)^{x + ]i + z) = m-\-n-\-p. 

The division is performed as follows : 

Separate the dividend into the three parts mx + my-fmz, 
n X 4- n y + n 2, and px + py-\-pz. The first term of the (quotient, 
m, is found by dividing m x, the first term of the dividend, by x, the 
first term of the <U visor ; multiplying the entire divisor by m will 
produce the Jirst part of the dividend. The second term n of the 
quotient is found by dividing the first term of the second part of the 
dividend by the first term of the divisor ; multiplying the entire di- 
visor by n will produce the second part of the dividend. The third 
term p of the quotient is found by dividing the first term of the third 
part of the dividend by the first term h{ the divisor ; multiplying the 
entire divisor by p will produce the third part of the dividend. The 
work is conveniently arranged as follows ; 

5 



( ^MV£RS(Ty\ 



66 



ELEMENTS OF ALGEBRA. 



g 










+ 










g 










■v..^ 










w 


M 




f M 


N 


^i 


fts 




a, 


^1 


4- 


+ 




4- 


+ 


?=^ 


5>i 




^ 


5ss 


a^ 


a, 




a, 


a^ 


+ 


+ 




+ 


+ 


H 


« 




H 


H 


a. 


^H 




^i 


a, 


4- 


+ 








M 


l^ 


?\i 






$ 


8 


s 






+ 


+ 


+ 






?s» 


5rj 


?s^ 






S 


S 


S 






+ 


+ 


+ 






« 


H 


H 






s 


s 


S 






4- 










N N 










fc s 










+ + 










>i a>i 










S g 










+ + 










« « 










1. ^ 










N 




^*" 






+ ^- 










>. -g 




•+3 




d 

■^ 

d 


quotie 




O 




'S 


-l 


a 


^^"^ 


"o 


1 


f 


1 




1 


-•^ 


^ 


o 


•S 


^, 




^3 


o 


^ 


05 


'^ 


^ 




^ 


^ 




'^u 


c 


^ 


si 


i=l 




d 


a 


^ 


rt 


13 


3 


f-i 


rt 


;-• 


F-i 




o 


o 


d 


O 






w 


Tl 






•iH 


'eS 




^ 


% 


_> 


'> 


'S 


§ 


'S 


OJ 


^ 


O) 




(B 


2h 


O) 


^^ 




_t- 


J 




c 


"3 


rt 


d 


O) 


cj 


<u 




(u 


%-i 


■^ 


V-( 


n:J 


^_i 


o 


B 


o 


d 

c3 


o 


-u 


S 


-M 




-u * 


o 


o 


"73 


o 


:=i 




|3 


rt 


d 




-ij 

g 


^ 




n:) 


o 


o 


Q 


o 






&4 




;h 


PU 


S 


fl^ 


m 


PU 



+ 

+ 
CO 

I 
I 



^ I 

I 7 

I 

CO 

I 



+ + 

CM 



H 


H 


1 


CO 
1 


1 


1 


1— 1 




4- 




H 




(M 




4- 




1i 




CO 

1 




1 
1* 





1 




1 


1 


1 


« 


1 


1 


CO 
1 


1 


^ 


1 


1 


1 


1 


1 


(?q 


-t 


CO 


CO 


+ 


1 


4- 


4- 


"« 


TO 






CO 


CO 






+ 


+ 






Tj 




^ 


t, 


(M 




(M 


1 


i 
1. 


1. 


1 


1 


. CM 

1 


1 











fj" 










o 






.2 
a> 




f 

^ 

J 


1 


m" 

*> 
^ 




d 


0) 


.^ 


d 




•5 


1 


"-I3 
d 

0) 




d 

■^3 


% 


^ 


^ 


+3 


S 


S 


i 
1 


(N 


CO 


•1-1 








1^ 


ptH 


1 


OQ 


1 



ALGEBRAIC DIVISION. 67 

Explanation Dividing the first term of the dividend by the first 
term of the divisor, we have a:*, the first term of the quotient. Now 
as we are to find how many times x* — 3a:^-j-2z+lis contained in 
the dividend, and have found that it is contained a:* times, we may 
take X* time.s the divisor out of the dividend, and then proceed to find 
how many times the divisor is contained in the i-emainder of the divi- 
dend. Dividing the first term of the remainder by the first term of 
the divisor, we have — 2 a:, the second term of the quotient. Simi- 
larly, we find the third term of the quotient. Hence, the quotient is 
x« - 2 X - 2. 

Notes: 1. Algebraic division is strictly analogous to *Mong division*' in 
Arithmetic. The arrangement of the terms corresponding to the order of suc- 
cession of the thousands, hundreds, tens, units, etc., and the processes for both 
are exactly the same. 

2. It is convenient to arrange both dividend and divisor according to poioers 
of the same letter ascending or descending. 

3. It may happen the division cannot he exactly performed ; we then alge- 
biaicaUy add to the quotient the fraction whose numerator is the remainder, 
and whose denominator is the divisor. Thus, if we divide x"^ — 2xy —i^ by 
X — y, we shall obtain x — y in the quotient, and there vnll be a remainder 

— 2ya. Hence, (xS - 2xy - y^) -r (x - y) = x - y - ~~. ■ 

X y 

Example 2. Divide a* + 6' -I- c» - 3 a 6 c by a + 6 -f- c. 
Arranging acconling to the descending powers of a, we have: 
Process. Dirisor. Diridend. Quotient 

a-f6+c)a» -3a6c-|-6»H-c»(a*-<26-ac 

a* times the divisor, a»-t-a^fe-fa^ 

First remainder, -aV)-a*c -3a6c+6»4-c» 

— ab times the divisor, —aV* — ab^ — abc 

Second remainder, -a^(H-ab^ -2a^c+6»+c» 

— ac times the divisor, — q^c —ac^— ahc 

Third remainder, ah^^ac^ abc-{-l^-\-c* 

b^ times the divisor, ab^ ■ -\-b^-\-b^c 

Fourth remainder, ac*- abc-b^c+c* 

c* times the divi.sor, ac' -^bc^-c* 

Fifth and last remainder, -abc-b^c-bc^ 

— be times the divisor, —abc—b^c-bc^ 

To verify the work, multiply the quotient by the divisor. 



68 ELEMENTS OF ALGEBRA. 

Example 3. Divide i^ xy^ + \x^ + ^y^ by ^y + ^x. 

Process. 

^x + ^y)\x^ +^xy^+^y^{j^x^-^xy + {y 

Divisor X ^x\ ^x^ + ^x^y 

First remainder, — i ^^2/ + -^ ^ 2/^ + i^y^ 

Divisor X — ^xy, — ^x'^y — ^ x y^ 

Second and last remainder, i ^ ^^ + iV 2/* 

Divisor X \ y% i ^ J/" + i^ y^ - 

Hence, in general, 

To Divide a Polynomial by a Poljmomial. Divide tJie first 
term of the dividend hy the first term of the divisor for the 
first term of the quotient; multiply the entire divisor hy this 
term, and subtract the product from the dividend. Divide 
as before, and repeat the process until the work is completed. 

Exercise 26. 

Divide : 

1. 14 x^ + 45 ^2/ + "^8 x^i/ + ^bxf +14.y^ hy 2x^ 

+ 5 xy + 7 y\ 

2. x'^ — 2x^y+2x^y^ — xy^ by x — y; a^ — 2ah^-^h^ 
hj a-b. ^ 

3. f-5y^ + 9 2/-6y^-y+2 by y^ - 3 y + 2 ; 
y^-1 hy y-1. 

4. x"^ + xy + 2 xz—2y^+ 7 yz — 3 z^ hy x — y +3z; 
c^ —b^ by a — h. 

5. 2^2/+ 36?/ + 10fe;:c + 15 62 by 2/ + 5&; a6 + a^b 
hy a -\-b. 

6. .125ic3-2.25a;22^4. \Z,^ xy^ -27 y'^ by .^x-3y, 

7. ?/-62/^-2^ + 54^-3a;2?/ by 2x-y\ x^-y^ by x + y. 



ALGEBRAIC DIVISION. 69 

8. a^t/^—a^ — y^+lhyxy + x-\-i/+l;4:i/ + 4:y 
-y8 by 3y + 2i/2 4-2. 

9. s^+i^ — z^-{-3xyzhyx + y — z; x—y by x^ — yi. 

10. 3^y^-\-2xi/^Z'-a^z^-{-i/^z^ by xy + xz + yz; 
x^ — y^ hy xi — yi. 

11. 12x*-26a^y-Sx^f+10xf-Sy^hySx^ 
-2xy-\-f. 

12. a^^'f^-Zxy-\hyx-\-y-\) 5V ^ " iV ^ 
+ iV ^ - S*I by J X- - i. 

13. 12 ;c« ?/9 - 14 x* 2/^ + 6 a;2 ^9 _ ^9 by 2x^y^-f) 
a^ — y^ by a;« — y^. 

14. a^ 6 - a ^ by a3 + 63 + a ^2 + ^2 J . ^ ^ ^4 _ -^ 
— x~^ by a; — x~^. 

15. a^ + :r* y + a:^ 2/* + i^ ?/^ 4- a: ?/* + 7^5 by a;^ + 7/^ ; 
al — 6f by a^ — fci 

16. Ja3 + Y«^-l-25a + 2.25 by Ja + 3; .Ibx^y^ 
+ .048 a:^ by .2 a;^ ^ 5^,^ 

17. at-a2-4at + Ga-2ai by J-4«i + 2; a:S_y5 
by a; — ;/. 

18. a;8 + 7^ + 23 4. 3 ^3y ^. 3 aj^J by a; + y + 2 ; .5 a;8 
+ a.^+ .375 a: + 75 by J a; + 1. 

19. x^-\-^y^^-^-^xyzhy 3^-\-^f^7?-xz-2xy 
-2yz. 

20. ^g a:* - I a,-3 - J r^ + I a: + Jg^ by 1.5 a:^ _ 3, _ | 

21. ofi — 1^ hy a^ + Qc^ y -\- X y^ ■\- ij^ , a^ — 7^ by a^ 
+ a; y + 2/2. 



70 ELEMENTS OF ALGEBRA. 

22. 10 a^--27aH + 34:a^I^-lSah^-8b^hj 5 a^-6ab 
- 2R 

23. 36x^+^i/+.25-4:X7j-6x + ^y by Qx-^-.d. 
24 ai2 + 2 aH^ + Z^^^ by «* + 2 ^2 ^2 + ^,4. ^6_2,6 ^y 

a^-2a^b + 2ah^-h\ 

25. 2 ^^" — 6 iz2« y- + 6 rr"2/2" - 2 ?/3« ^y 2^ — 2/«; ^» 
+ 2/3" by it" + ?/. 

26. i6'2«— 7/2'" + 2 7/"*^'- s2' by ^''+ ?/"• — ;3'; 32^-22^ 
by 3^ - 2". 

27. ifX^-lil-xi/^ by f^-.752/; a:-t"'-3aj-i"'2/-i'' 
+ 2 y-h by a- ^"^ — 2/-i". 

28. ?/2a;2m^2 7/22:'"+"+ 2 2/r a?"* + 22a;2n + 2ra?";2 + r2 
by 3/ a:*" + 2; ^" + 7\ 

29. a;~i + 2x~^y~i + 2/"^ by a?~^ + y~^; x^ + ?/* by 
a?2 + 22 a? 2/ + 2/2, 

30. x~'^ — y'~'^-\-2y~^z~^ — z~'^ by a;~i + 2/~^~^~^j 
a?* — 3 2/* by x — y. 

34. There are special methods for finding the quotient 
of binomials, hy inspection, which are of importance on ac- 
count of their frequent occurrence in algebraic operations. 
Thus, 

It may be shown by actual division that : 

a—b ^a—b 

^^—^ = a^+as 6+a2 h^+ab^+b^ ; ^"—^ = a^ + a'^b + a%'^ + 02^8 +ab^+b^) 
a—b a—b 

and so on. Hence, in general, it will be found that, 



ALGEBRAIC DIVISION. 71 

The difference of any two equal powers of two numbers is 
divisible by the difference of the numbers. 

In each of the above quotients we observe the following 
laws : 

I. The number of terms is equal to the exponent of the 
powers. 

II. Hie signs are all positive. 

III. The exponent of a in the first term is one less than 
the exponent of a in the first term of the dividend, and in 
each succeeding term it decreases by one {in the last term its 
exponent is 0, or a. disappears). 

The exponent of b in the second term is one, and in each 
succeeding term it increases by one {in the last term its expo- 
nent is one less than the exponent of b in the dividend). 

IV. The first term is found by dividing the first ter^n of 
the dividend by tJie first term of the divisor. 

V. To find each succeeding term, divide the preceding 
term by the first term of the divisor, and multiply the restUt 
by the second term of the divisor regardless of sign. 

Example. Divide 1 — w* by 1 — n. 

Solution. Dividing 1, the first term of the dividend, by 1, the 
first term of the divisor, we get 1 for the first term of the quotient 
Now divide the first term of the quotient by the first term of the 
divisor, and multiply the result by n, the second term of the divisor 
(regardless of sign), for the second term, n, of the quotient. Dividing 
the second term of the quotient by the first term of the divisor, and 
multiplying the result by n, we have n^ for the third term of the quo- 
tient. Similarly, we find n*, and n* for the fourth and ffih terms, 
respectively. .*. (1 — n*) -r (1 — n3 = 1 -|- n -f n* + n* + n*. 



72 ELEMENTS OF ALGEBRA. 

Exercise 27. 

Divide by inspection : 

1. m^ — n^ by m — n\ a^ m^ — h^n^ by am — hn\ 
m^n^ —1 by mn — 1. 

2. l—m^n^a^hjl—mnx-jix yj*— {x zj hy xy — xz; 
1 — a''b'' x'^ hj 1 — ahx. 

In order to apply this principle the terms of the divi- 
dend must be the same powers of the respective terms of 
the divisor. It is not necessary that the exponents of the 
terms of the divisor be 1, nor that they be the same, nor 
that the exponents of the terms of the dividend be the 
same. Thus, 

Example Divide x^^ — y^^ by x^ — y*. 

iSolution Dividing x^^ by x^, we have x^ for the ^rst term in the 
quotient. Now divide x^ hj x^ and multiply the result by y*, for the 
second term, x^ y*, in the quotient. In like manner we find x^y% and 
y^^ for the third and fourth teims of the quotient. 

. •. (a;i2 - i/16) -i- (x3 - 2/4) = a;9 + ^6 ?y4 + a:3 2/8 + ^12 

So in general x"* — y"^ divides 2:"" — ;v""* {n being any 
positive integer), since the dividend is the difference be- 
tween the nVa powers of the terms of the divisor. 

3. a^ - W by a^-l^- x^^ - f^ by x^ - y'^ \ x^^ - y^'^ 
by a::^ — y'^. 

4 ai5 - &30 ]3y a3 - &6 . x^m _ ^35n -^^ ^n _ ^n . 2io« 

- a^"" by 22" - x"^. 

We may easily apply these principles to examples con- 
taining coefficients as well as exponents; also to those 
involving fractional or negative exponents. Thus, 



ALGEBRAIC DIVISION. 73 

Example. Divide 81 a^^ - 16 A** by 3 a« - 2 h\ 

Solution. Dividing 81 a" by 3 a«, we have 27 a» for the frst 
term of the quotient. Now divide 27 a* by 3 a* and multiply the 
result by 2 6*, for the second term, 18 a® 6*, in the quotient. Simi- 
larly, we find 12 a' 6^*, and 8 h^^ for the <At>d and fourth terms in the 
quotient. 
.-. (81a"-16634)^(3a»-26«) = 27a»+18a«6«+12a«6»2+86". 

If a and h are coefficients, a^a^** ~ 6"^*"" is divisible by 

ax' — hif^f since the dividend is the difference between 

^1 -1 
the Tith powers of ax* and ////'". In general, a? "•— y *" 

na n$ 

divides x '^ — y "" (n being any positive integer), since 

a 

the latter is the difference between the ?ith powers of a; •* 
and y '. 

5. 64ai2_27?i» by 4a^-Sn^; 16 2^ y^^"* - ^ m^z^* 
by 4a:8y6*_ j^^4 2io» 

6. «8 a^S" - 68 2/3« by a2aJJi' _ ^^ ^^^ . 32 ar^o - 243 y^ 
by 2 0^2 - 3 3/3. 

7. x~^ — y~i by a;~i — y~i ; 3^ — y^ by a;^ — yi ; 
a xi — 6i y by ai x^ — fci yi 

35. It may be shown by actual division that : 

ai—l)i a*-h* 

— r-j- = a — b; — —r- = a*—a^b-{-ab^—b*; and so on. 
a+o a + o 

Hence, in general, it will be found that, 

The difference of any two equal even powers of two num- 
bers is divisible by the sum of the numbers. 

In each of the above quotients we observe the laws are 
the same as in I. and III., Art. 34 ; also, 



74 ELEMENTS OP ALGEBRA. 

VI. The signs are alternately + and — . 

Hence, the principle may be applied to different classes 
of examples as in Art. 34. Thus, in general, 

If a and l are coefficients, ax^ -{■ h'lf divides a" x"*^ 
— If y^"^ (n being any even and positive integer ; also m and 
p may be integral, fractional, or negative), since the divi- 
dend is the difference between the nth. powers of a x^ and 
by-. 

Note, 'rtie difference of the squai'es of two numbers is always divisible by 
the sum and also by the difference of the numbers. Thus, 06 — 68 jg divisible 
by aS ± 64. jn general, a^n — IZm [§ divisible by a« ± **" when n and m are 
integral. This is the converse of Art. 26. 

Exercise 28. 

Divide by inspection : 

1. 625 a^x^ — 81 ra^n^ by 5ax+ 3 mn; a^ — h^ hy 
x:^ + b^; x^ — 1 by x + 1. 

2. x^ - yi by x^ + y^ ; 256 ir* - 10000 by 4£c + 10^, 
3 ^im^yin by a;"* + 2/" ; tV ^^ - -^^^^ 2^^ by J rr^ + .22/1 ; 

^10 _ ^,10 ]^,y a, + 6. 

4. 729 ai2 _ 64 ^,18 ^y S a^ + 2 b^ ; a^ x^"" - ¥y^"* by 

a ^-^ + &2 y2 m 

5. a""^ — x'"^ by a~^ + a;"^; a^ x~^ — b^y~^ by 
a a:~3 + h's y~^- • 

6. xh - yi"" by 2;i*^ + t/t^"*; 81a-i"a; - Jq &f«^-f'» 
by 3 a-T5'*a;4 + | b^y'^'^. 



ALGEBRAIC DIVISION. 75 

36. It may be shown by actual division that : 

j- = a^—ab + b^', j- = a*—a*b+a^b^—ab^-^b*; and so on. 

a-\-b a + b 

Hence, in general, it will be found that, 

The sum of any two equal odd powers of two numbers is 
divisible by the sum of the numbers. 

In each of the above quotients we observe that the laws 
are the same as in Art. 35. 

Hence, the principle may be applied to all the different 
classes of examples as in Art. 34. Thus, in general, 

If a and b are coefficients, aaf-^-hy"^ divides a" a:"' 
_j_ i^ynm ^^^ beiug ani/ odd and positive integer, also m and 
p are integral, fractional, or negative), since the dividend 
is the sum of the nth powers of aaf and by^. 

Exercise 29. 

Divide by insj^ection : 

1. x" ■\- If by X -\- y \ x~^ + y~^ by a;"^ + 3rM 
1024 a:« + 243 y« by 4 a; + 3 y. 

2. 128 x^^ + 2187 y" by 2 2:3 + 3 y2 . 243 x^^ + 32 ^o 
by 3 a:8 4. 2 y2. 

3. a;!*" + 2/21*" by 3^^ ■¥ y^^ - A:2i« 4- 7^86- ^y l^^^m^''', 
a" + 6" by a + 6. 

4. wi w- + a; y by wi ?ii + xi yi ; x^ + ?/V by x^ -\~ yi\ 
^\-\- y'^ by a;-i + y-K 

5. ax\^b^yhya\x\^biiy\', (# -f ( J) by (f )i + (f )i ; 
a^ + 6^0 by a + 62. 

Note. Since a« and Ifi are odd powers of a^ and h^, n^ + 6« is divisible by 
cfl + 62. aW and ft" are the 5th powers of a^ and 62, a 10 + 6W is divisible by 
a* + 6^. Also, a» and 6* are the third powers of a« and 6«. Therefore, a« -f 6» 
is divisible by a* + b^. 



76 ELEMENTS OF ALGEBRA. 

6. a^ + 612 by a* + &^ 2:6 + 1 ]3y x^ +,1 ', x^^ + 1 by 

a;* + 1 ; a27 + ^27 by a^ + h^. 

7. a;io + 2/10 by 2;2 + 2/2 ; a;!^ + 2/^^ by a^ + y^ ; 64: + x^ 
by 22 4- 2;2 

8. 64 x^ + 729 f by 22 a;2 + 9 ^^2 . ^lo + __l^ by x^ + ( J)2; 

^24 4. 524 by «§ + &8 ■ 

9. ai8 + Z,i8 by a^ + b^ and by a^+P; j^^ x^ + J^ f 
by i^^ + i2/^- 

10. ^36 + J36 by ai2 + ^,12 and by a^ + J*; 729 a;^ + 1 
by 9 a:2 + 1. 

11. a;*2 + 2/42 by x^ + ^6 and by x^^ + 2/^*. Query. Is 
it divisible by 2;2 + / ? Why ? a^^ + &18 by a^ + h^ ', 
^27 + J21 by a9 + &' ; a^^ + 515"' by a^ + J^. 

12. rr^ + 2/^* by o^^^ + ?/i^ and by x^ + /. Query. Is 
it divisible by 2^2 + 2/^ ? Why ? a^ + &12 by a^ _^ &* ; 
aH9 + m27n36 by a^h^ + 77i9 7ii2; ^15^25 _^ m^o^iio by 
a^ 6^ + m^ 7i2 

Find an exact divisor and the quotient for each of the 
following, by inspection : 

13. 8 + a^; ^6 - 6^ 8 - ^; a:* - 81; a^^ - h^^; 
81 ai2 - 16 h^; a"^- 625 ; a^ - h^ 

14. aj20 - 2/15; ^,^5 + ^5. ^12 _ ^12 . ^6 _ 1 . ^-12 _ 2,-12. 
a^a^^-h^fp- a^-'62) 16 «* - 81. 

15. Z2w> - h^; 81 «8 - 16 b^; 1 - y'^ ; a^ x^ + 1000; 
a* a;* — 1 ; a^ + m^a:^ ; 2^2 7/2 — 81 (x2 

16. 32 ^10 - 243 2>i5. cIo.tIO'* - a^^x^^; a^" + ftQ''^ 
a^x^ + b^y^""; c^ x^^ + 6' 2/^". 



ALGEBRAIC DIVISION. 77 

17. aj-^" + y-e*; 8a^y3 4. 729; fti2 yi2j» _ Ji:^yi2». 
cS^^Sp _ j8^«^ 2:4 _ 1296; ^^S" - ftio* 

18. 128.^21 + 2187^^*; 256a;i2-81 //«; a6»65* + a:S'y^'; 
1 + 128 a:!*; «-«'»- d^^-. ^j-ii^-i-l. 

19. x^-y-i_aiy-i; a-i^-S+l; i^a^x^'^^^^h-'^f^ 
g^ x-t"-. 00032 y-^". 

20. ii c" + .002432;2/-i; 256 aa;-t~ - .0081 J-f*; 

Queries. How divide a monomial by a monomial ? Prove it. 
How prove the method lor dividing a polynomial by a polynomial \ 
In Art. 35, the sign of the last term ot" the quotient is — , while in 
Art. 36, the sign ot* the last term of the quotient is 4-. Why is this ? 
What is the product of a* and a- * ? Prove it. What of a* and or* I 
Prove it. 

Miscellaneous Exercise 30. 
Divide : 

1. a3 6-3+ T^a2 6-2 + ^1 by Jafe-a + JJ-i; ari + y-i 
by x~^ + ir^- 

2. a-6+ 5a-H-i+10ft-36-2-f i0a-2 6-3 + bcrH'^ 
+ 6-6 by a-i + b-\ 

3. 2 a2 - a^ - 2 rt + 1 by 1 - a^ ; x - ij hy x\ - iji. 

4. (a — 6 - c)" - (a - 6 - c)"- "• - (rt — 6 — c)"" by 
(a- h-cyr 

5. 2 2^3 + 2 y3 + 2z^- 6zyzhy (x - yf -\- (y - zf 
+ (z-2:)2; (2^-ff by {x' + xy + ff. 

6. {x^-2yzf-^f7? by ^^2-4^2; (a:+2y)8 + (y-3«)3 
by ar + 3 (y - z). 



78 ELEMENTS OF ALGEBRA. 

7. x^ — x^i/ -\- xy^ — 2 xi y^ + y^ by xi — xy^ -\- x^y 
-yl 

8. 2 :r3« _ 6 ,jc2nyn _j_ e ^n^2n _ 2 fn ^^ ^n _ ^n^ 

by :r"' - 3 a;'""^ / - 6 a:'"-^ ?/2«. 

10. a3" - 3 a2« 5« + 3 a« &2« _ 53^ by ^« _ j« . 3 ^-o: 

- 8 a^ + 5 ^3^ - 3 a-3^ by 5 a^ - 3 ^-^ 

11. 6 «'« + ' - 23 a^" + ^ + 18 a^ - a^"-^ - 3 a'"~' 
4. 4a3«-3 - a""-' by 2 a^''^^ - 5 a^" - 2a2'*-^ + a''-'. 

12. 4 at - 8 ai - 5 + 10 ^-^ + 3 a"! by 2 ai2 - aiV 

- 3 h~l 

13. 6:r^+3-5:2r" + '-6 2r^ + ^+19af-21r^-^ +4,r^-2 
by 2 x^ + x^ — 4:X. 

14. 6m*-"-*-2 + m*-"+^ - 22 m^"" + 19 7?t*-"-^ 

15. 6aj^ + "+'+2f^-"-^^-9af + "+lla;^ + "-'-6;zf + "-2 
^^ + n-3 by 2 2^'^-^'+ 3a;" + ^-a;". 

16. a""* — ct" fe(''-i)'» — a<"*-i>" &"* + 5"*" by a" - &**. 

Find an exact divisor and the quotient of the following, 
by inspection : 

17. 8a^+l; 16 - 81 a^ ; 64 a^ - 8 &3 . ^34. iqOO ; 
^6 - 64 ; m^ -71^; 1 - 8 ?/3 ; a^ 6^ - 1. 

19. 8 ;x^6 - 27 ?/-^ 64 ai2 _ 27 ^-9; 243 a^ + 32; 

cSjc^'* - ftS^Sm . 1^ ^4n _ 0016 hh'^', 29" al2» + 36n 



EVOLUTION. 79 



CHAPTER VIL 
EVOLUTION. 

37. Evolution is the operation of finding one of the 
equal factors of a number or expression. Evolution is the 
inverse of involution. 

By Art. 27, (2 a)» = 4 a^; (2 «)» = 8 a* ; (2 a)* =16 a*; etc. 

2 a is called the secoml or sqiuire root of 4 o^ l)ecause it is one 
of the two ec[ual factors ot 4 a'^; it is the thii-d or cube root of 8 a* 
because it is one of the three equal factors of 8a^; etc. Hence, in 
general, 

A Root is one of the equal factors of the number or 
expression. 

Roots are indicated by means of fractional exponents, 
the denominators of which show the root to be taken. 

Thus, (a)* means the second or square root of a ; (a)* means the 
third or cube root of a; (a*)* means the sixth root of a^. In general, 
(a*")* means the nth root of a"*. 

Roots are also indicated by means of the root sign, or 
radical sign, ^. 

Thus, \/a means the square root of a ; ^a means the cube root 
of a ; v^ means the nth root of a"». 

The Index is the number written ia the opening of the 
radical sign to show wliat root is sought, and corresponds 
to the denominator of the fractional exponent. When no 
index is written, the square root la understood. 



80 ELEMENTS OF ALGEBRA. 

« — 1 
Note, ya or a» is defined, when n is a positive integer, as one of the n 

equal factors of a ; so that if Va be taken n times as a factor, the resulting 

product is a ; that is, ( ya)'^ or i^a" j" = a. 

,mn -\mn ( -^\mn, 
Similarly, ( \/a) or Va*""/ = a, 

38. The sign, ± or =p, is sometimes used and is called the double 
sign; it indicates that we may take either the sign + or the sign — . 
Thus, a ± 6 is read a plus or minus b. 

By Art. 27, (+a)4 = a^ (-a)4 = a4; (+a)5 = a5; (-ay = -aK 
Therefore, (a*)^ = ±a; (+ a^)^ = a; (— a^)^ = — a. Hence, in 
general, 

Hven roots of any nu7nber are either positive or negative. 

Odd roots of a nwniber kave the same sign as the number 
itself. 

Since no even power of a number can be negative, it 
follows that, 

An even root of a negative number is impossible. 

Such roots can only be indicated, and are called imaginary. Thus, 
(— a^)^y Y^— 6, Y^— 1, and ^— a^, are imaginary. 

Example 1. Find the square root of 9 a^b^c^. 

Solution. Since, to square a monomial, we multiply the expo- 
.nent of each factor by 2, to extract the square root we must divide 
the exponent of each factor by 2. The two equal factors of 9 are 
3 X 3, or 32. Dividing the exponent of each factor by 2, we have 
3 a^ b'^ c. Since the even root of a positive number is either positive 
or negative, the sign of the root is either plus or minus. 
.-. y/9aH^ = ±3a^b^c. 

Example 2. Find the fifth root of - 32 a^^ x"*. 

Solution. Since, to raise a monomial to the fifth power, we mul- 
tiply the exponent of each factor by 5, to extract the fifth root we 
must divide the exponent of each factor by 5. The equal factors of 
32 are 2 X 2 X 2 X 2 X 2, or 2^. Dividing the exponent of each 



EVOLUTION. 81 

factor by 6, we have 2 a* x*. Since the odd roots of a number have 
the same sign a.s the number itself, the sign of the rout is minus. 

.♦. J^— 32 a*® x*» = — 2 a* a*. Hence, in general, 

To find any Boot of a Monomial Resolve ike numerical 
coejffkient into its prime factors, each factor being written 
with its highest exponent, divide the exponent of each factor 
by the index of the required root, and take the ^product of 
the resulting factors. Give to every even root of a positive 
expression tlie sign ± , and to every odd root of any expres- 
sion the sign of the expression itself 



Hote. Any root of a fraction is found by taking the required root of each 
V27 



of ito terms. Thus, .Vl _ J^ = 2 in general, t/- = -^ 



Exercise 31. 
Find the value of the following expressions : 

1. V25^; ^-%aH^a^', y/-12baH^', v^81ai«W 

2. (-343a*5-6)i; {\{7^ifz^)^; {-x^^y^^)^- v'Sl^V^. 

3. v^iw. (I21a:i2y2)i. V25 aH"^-, (16a-8&8)i 

4. (- 243 a^* Jio«)i. (_ 54 ,,,3 ^6 ^^J. (^w ^80)tV. 

5. (-32ai0y-5)i; V^2W^r^ -, (625 a^ 6i« c*)i. 

6. (512an8ci5rf-8)i; V64a-«&"*; -y/m ; ^/- 32 ai^. 

8. V'i^^; (2"a2«54n2^)i; V8l2:5i-y2m+4. >^_ 8 a;8*-«ye-+». 

A 



82 * ELEMENTS OF ALGEBRA. 

10. \/l6x^''y^''z^; (- iV^ ^^- ^ ?i" ^M ; V^-^^Sm^-m 
Simplify : 

11. Sj^a^hU-l + iij a^ hh d)^ - (f i a* ol c-^)i 

— V ^^ <^^« c 4. 
V 50 

Express the nth roots of: 

12. 3x7x4; 52:"2/2^ 3a^&3. (a-ir)3; (^^z)"; ^'"-y"; 

Express by means of exponents : 

13. \/JWc^; 7a(x-yT; V^^^; V^(^ + 2/)~ 

Queries. If n and j9 in the last two parts of Ex. 13 are integral, 
what signs should the roots have ? Why ? When should the first 
two roots have the double sign ? 

39. By Art. 28, (a + hy = a^ + 2ah -\- b^. 

Therefore, (a^ + 2 a 6 + 62)i = a-^b. 

By observing the manner in which a + b may be obtained from 
a^ + 2ab + b^, we shall be led to a general method for finding the 
square root of any polynomial. 

Process. a"^ + 2ab -h b^(a + b 

First term of the root squared, a^ 

First remainder, 2 ab + b^ 

Trial divisor, 2 a 

Complete divisor, 2a + b 



Complete divisor X b, 2ab + b^ 

Explanation. The square root of the first term is a, which is 

the first term of the required root. Subtracting its square from the 

given expression, the remainder is 2ab + b^, or b times 2 a -\- b. 



EVOLUTION. 83 

Since the first terra of the i-einainder is twice the product of the fii-st 
and last terras of the root, and we have found the first terra ; there- 
fore, divide 2 a6 by twice the fii-st terra of the root already found, or 
2 a. 'I'he result will be the second terra b of the required root. 
Adding 6 to the trial divisor gives the complete divisor, 2 a + 6. 
Multiplying by 6 and subtracting, there is no remainder. 

By Art 28, (a-\-b+cy = a^+2ab-{-b^+2ac + 2bc + c^. 

Therefore, (a^-{-2ab-^b^-\-2ac-^2bc + c^^ = a + b+c. 

Process. a^-\-2ab-\-b^+2ac-^2bc + c'^{a-\-b + c 

First terra of root squared, a^ 

First remainder, 2ab + b^-{-2ac + 2bc+c^ 

First trial divisor, 2 a 

First complete divisor, 2a-\-b 



First complete divisor X 6, 2ab-{-b^ 

Second remainder, 2ac-\-2bc-^c* 

Second trial divisor, 2a-|-26 I 

Second complete divisor, 2a-\-2b-i-c \ 

Second complete divisor X c, 2ac+2bc-\-c^ 

Xtxplanation. Proceeding as before, the first two terms of the 
root are found to be a + 6. To find the last terra of the root, take 
twice the terms of the root already found for the second trial divisor. 
Dividing 2 a c by the first terra, the result c will be the third term of 
the required root. Adding this to the trial divisor, gives the entire 
divi.<*or. Multiplying by c and subtracting there is no remainder. 
We have actually squared the root and subtracted the square from 
the given expression. Hence, in general, 

To find the Square Root of any Polynomial Arrange the 
terms according to the powers of one letter. Find the square root 
of the first term. This will be the first term of the required root. 
Subtract its sciuare from the given expression. Divide the first term 
of the remainder by twice the root already found. The quotient 
will be the next term of the root. Add the quotient to the divisor. 
Multiply the complete divisor by this terra of the root, and subtract 
the product frora the remainder. For the next trial divisor, take 
two times the terms of the root already found. Continue in this 
manner until there i-* no remainder. 



84 



ELEMENTS OF ALGEBRA. 



Example. Find the square root of 4 a — 10 a^ + a^ + 4 a^ + 1. 
Arranging according to the ascending powers of a, we have, 

[_2a2-a3 



Process. 

First term of root squared, 
First remainder, 
First trial divisor, 2 

First complete divisoi' , 2+2a 



2 a times first complete divisor, 
Second remainder, 
Second trial divisor, 2+4 a 
Second complete divisor, 2+4 a— 2 a^ 



1+4 a -10a3+4a5+a6(i4-2a 

_1 

4a-10a3+4a5+a6 



4a^+4q 

-4a'-^-10a3+4a5+a« 



— 2a^ times second complete divisor, 
Third remainder, 

Third trial divisor, 2+4 a— 4 a^ 

Third complete divisor, 2+4 a— 4 a^—cfi 



-4a2-8a8+4a^ 
-2a8-4a4+4a6-(-a« 



a* times third complete divisor. 



~2a3-4a4+4a5+a« 



Note. The student should notice that the sum of the several subtrahends 
is the square of the root, and that he has actually squared the root and sub- 
tracted the square from the given expression. 

Exercise 32. 

Find the square roots of: 

1. 2^-4^3+ (]y2_4y4.i. 9^4- 12 a3_ 2^2+4^ + 1. 

2. 4a6_ l2a^l-~ 11 a^^ly^ + b^ a^ h^ - 17 aH^- 70 ab^ 
+ 49 &6. 

3. x^ - 12 a;5 + 60 x^ - 160 a^ + 240 x'^ - 192 a; + 64. 

4. 8 a + 4 + a^ + 4 ^3 +- 8 a2 

5. 9 +- a;6 4- 30 :r - 4 a^ + 13 x^ + Ux^ - Ux^, 

6. 6a62c_4a2&c + ^252 + 4ft2c2 + 9 52^2 - 12 a.&c2 

7. 49 a^ - 28 a^ - 17 a^ + 6 « 4- f. 



EVOLUTION. 85 

8. 4x^ + 9 y^ + 25 cc^^ 12x1/ -SO ay- 20 ax, 

9. VL^ — 6 a 771^ + 15 a2 7n* — 20 a^ vi^ + 15 a* m^ — 6a^7n 

10. 1 -2 a + 3 a2 - 4 a3 4- 5 rt* - 4 a5+ 3 a«— 2 a^ + a^. 

11. 9 vi^ — G 7« n + 30mx+6my-\-7i^—i0nx—2 ny 
+ 25r»+ lUa;^+ //2. 

12. 7^-\-l5x^{/^^- 15aj*//2 4. ,/6_ ^y _ 20xV- <^^"^y- 

13. 49x^/- 24x^/3 -30 ar8y + 25 a^+ 163^. 

14. a;6 - G 2:^ + 172:4 - 34 2:3 _^ 40 a:2 _ 40 2: + 35. 

15. 4 - IG «t + IG a^ + 12 rf - 24 at ?>i + 9 h. 

10. ^a:4_|^y^_^,,2,,2_,^y^^,^4. 2:4«_ 0a;3n+ 5^* 
4- 12 a;" + 4. 

17. 25 ^1 + 1 G - 30 X -24:xh + 49 .rf 

18. 9x-2+ 122;-V^-6a; + 42^-4ar^7/^ + ^. 

40. Since the square root of an expression is either + or — , the 
square root o[ a^ -\- '2 a h -\- h^ is either a + h or — a — h. In the 
process of finding the sfjuare root of a'* + 2 a 6 + />', we herein by tak- 
ing the square root of a*, and this is either + a or — a. If we take 
— a, and continue the work as in Art. 39, we get for the root -^a-^h. 
Also, the square root of a' — 2 a 6 4- />^ is either a — h ov — a -{- b. 
This is true for every even root. Hence, the signs of all the terms oj 
an even root may he changed^ and the number will still be the root oj the 
same expression. Thus, last process Art. 39, if — 1 he taken for 
the square root of 1 we shall arrive at the result — 1 — 2 a + 2 a'* 4- a*. 

41. Square Root of Numerical Numbers. The method 
for extracting the square root of arithmetical numbers is 
based upon the algebraic method. 



86 ELEMENTS OF ALGEBRA. 

Since the square root of 100 is 10, of 10000 is 100, etc., it fol- 
lows that the integral part of the square root of numbers less than 100 
has one figure, of numbers between 100 and 10000 two figures, and so 
on. Hence, 

If a point he placed over every second figure in any numher, begin- 
ning with units' place, the numher oj points ivill show the numher of 
figures in the square root. 

Thus, the square root of 324947 has three figures ; the square root 
of 441 has two figures. If the given number contains decimals, the 
number of decimal places in the square root will be one half as many 
as in the given number itself. Thus, if 2.39 be the square root, the 
number will be 5.Vi2i; if .239 be the root, the number will be 
6.057121 ; if 10.321 be the root, the number will be 106.523041. 
Hence, 

The numher of points to the left of the decimal point will shorn the 
numher of integral places in the root, and the numher of points to the 
right will show the numher of decimal places. 

Example 1. Find the square root of 45796. 

a +&+c = 214 
Process. 45796(200+10+4 = 214 

The square of a or 200, - 40000 

First remainder, 5796 

First trial divisor, 2 a, or 400 I 

First complete divisor, 2a-lh, or 410 [ 
First complete divisor X &, or 10, 4100 

Second remainder, 1696 

Second trial divisor, 2a+2&, or 420 i 

Second complete divisor , 2a+2&+c, or 424 | 
Second complete divisor X c, or 4, 1696 

Explanation. There will be three figures in the root. Let 
a -^h -\- c denote the root, a being the value of the number in the 
hundreds' place, h of that in the tens' place, and c the number in the 
units' place. 

Then a must be the greatest multiple of 100 whose square is less 
than 45796, this is 200. Subtract a^, or the square of 200 from the 
given number. Dividing the first remainder by 2 a, or 400, gives 10 



EVOLUTION. 



87 



for the value of b. Add this to 400, multiply the result by 10 and 
subtract. Dividing the second remainder by 2 a -f 2 6, or 420, gives 
4 for the value of c. Adding this to 420, multiplying and subtract- 
ing, there is no remainder. Hence, 214 is the required root; because 
we have actually squared it and subtracted this square from the 
given number and found no remainder. The student should observe 
that the .sum of the several subtrahends is the square of the root. 
Example 2. Find the square root of 17.3 to lour decimal places. 



Process. 

S<iuare of 4, 

First remainder, 

First trial divisor, 8 

First complete divisor, 81 

First complete divisor multiplied by 1, 

Second remainder. 

Second trial divisor, 82 

Second complete divisor, 825 



17.360<KK)06(4.1693..., 

16 

130 



81 
4900 



Second complete divisor nmltiplied by 5, 

Third remainder, 

Third trial divisor, 830 

Third complete divisor, 8309 



Third complete divisor multiplied by 9, 
Fourth remainder. 
Fourth trial divisor, 8318 I 

Fourth complete divisor, 83183 | 
Fourth complete divisor multiplied by 3, 
Fifth remainder, 



4125 
77500 



74781 
271900 



249549 
22351 



Let the student formulate a method for arithmetical square root 
from what has been demonstrated. 

NotM ; 1. If the trial divisor is not contained in the remainder, annex to 
the root, also to the divisor, then annex the next period and divide. 

2. Should it be found that after completing the trial divisor, it gives a pro- 
duct greater than the remainder, the quotient is too large, and a less quotient 
must be taken. 

3. Tf the last remainder is not a perfect square, annex periods of ciphers and 
procee<l as before. 

4. The square root of a fraction may be found by taking the square root of 
its terms, or by first reducing it to a decimal. 



88 ELEMENTS OF ALGEBRA. 

Exercise 33. 

Find the square roots of : 

1. 33124; 41.2164; Jf | ; ^%^^^; .099225; 1.170724. 

2. .30858025; 5687573056; 943042681. 
Find the square root to four decimal places of: 

3. .081; .9; .001; .144; if; .00028561; 3.25; 20.911. 

42. By Art. 29, (a + by = a^ + 3 a^ b + 3 ab^ + bK 

Therefore, (a^ + 3 a^ 6 + 3 a &2 + b^)l = a + b. 

By observing the manner in which a -\-b may be obtained from 
a^ + 3 a^ft + 3 a &2 -I- 6"^, we shall be led to a general method for find- 
ing the cube root of any compound expression. 

Process. a^+2a%yMb'^-\-b^ {a+b 

First term of the root cubed, o^ 

First remainder, 3a%+Zab'^-{-b^ 

Trial divisor, or 3 times the square of a, Za^ 

3 times the product of a and b, 2ab 

Second term of the root squared, 6^ 



Complete divisor, Za^+Zab+b'^ 

Complete divisor X &, Za%+Zab'^^-b^ 

Explanation. The cube root of the first term is a, which is the 
first term of the required root. Subtracting its cube from the given 
expression, the remainder hZaH + Zab'^+b'^, or b times Za^ + Zab+b^ 
Since the first term of the remainder is three times the product of the 
square of the first term of the root multiplied by the last term, divide 
Za^b by three times the square of the first term of the root already 
found. The result will be the second term 6 of the required root. 
Adding to the trial divisor three times the product of the first and 
second terms of the root, and the square of the second term, gives the 
complete divisor, ov Zw^^ Zab + b^. Multiplying by b and subtract- 
ing, there is no remainder. 

Since the cube of a+b-\-c is a^-\-Za'^b^-Zab'^-\-h^+Za^c + Qabc 
-f-3&2c + 3ac2+36cHc8, the cube root of a8 + 30^6 -|- 3 a 62+63+ 3 a^c 
+ 6a6c + 362c+3acH36c2+c« is a + b + c. 



EVOLUTION. 



89 



i 

CO 



4- 


^ 


^j 


o 


eo 


cS 


j; 




w 


^ 


1 


t 


s 


5 


-- 


^ 


^ 


^ 




t 


^ 


1 


•o 




e^ 


« 


$ 


# 



c o 



J 



cS 



<Sc^'" 



^5 ^ 



o 

I 



« ^ X 



^ 



iz 



•5 S ,2 

O C ;:2 



'tis -k.) 



I- 

Ph C 



■2 5 






-^ -r -^ -« -^ •> 

g ^ 3 B .5 -^ 

"^ q O) 4> eS ^_^ 

•2 iJ s S S is Ts 

£ -3 8 8 -2 '^ § 

£ I ^ ^ § §^ 



3,3. 

s a 

8 8 

o p 



90 ELEMENTS OF ALGEBRA. 

Explanation. Proceeding as before, the first two terms of the 
root are found to be a + h. To find the last term of the root, take 
three times the square of the terms of the root already found for the 
second trial divisor, and divide Za^c by the first term. The result 
will be the third term of the required root. Adding to the second 
trial divisor three times the product of a + & and c, and the square of 
c, gives the second complete divisor. Multiplying by c and subtract- 
ing, there is no remainder. Observe that the sum of the several sub- 
trahends is the cube of the root, and that we have actually cubed the 
root and subtracted the cube from the given expression. Hence, in 
general, 

To find the Cube Root of any Polynomial. Arrange the terms 
according to the powers of one letter. Find the cube root of the first 
term. This will be the first term of the required root. Subtract 
its cube from the given expression. Divide the first term of the 
remainder by three times the square of the root already found. The 
quotient will be the next term of the root. Add to the trial divisor 
three times the product of the first and second terms of the root, and 
the square of the second term. Multiply the complete divisor by 
this term of the root, and subtract the product from the remainder. 
For the next trial divisor, take three times the square of the root 
already found. Continue in this manner until there is no remainder 
or an approximate root found. 

A Term may be a figure, or a letter, or a combination of 
figures and letters, or of letters only, produced by multi- 
plication or division, or both. 

Thus, in the algebraic expression 5 + 2a®6* — a+ -gi^; 5, 2a^b*, a, 
ah- "^y 



«2 ,,w 



are terms. 



An Algebraic Expression is a representation of a number 
by any combination of algebraic symbols. 

Example. Find the cube root of 27 a — 8 a^- 36+ 36 a^- 12a- ^ 
-54a^ + 9a-§+27a§ + a-6-6a-J. 

The work is conveniently arranged as follows : 



EVOLUTION. 



91 



+ 


I 


1 


+ 


+ 


•4M 


•* 


HB« 


o 


l„ 


1 


<N 


C3 


o 


J 


CO 


«o 


CO 


1 
1 


1 

1 


**— ^ 


^ 


e 




C5 


05 




4- 


+ 




r* 






k 

1 


I 

1— • 




1 

CO 

1 


1 




o 


1 




t^ 


r^ 




04 


(N 




+ 


+ 




■an 


im 




c2 

1 


<2 

1 




1 


1 




5 


^ 




CO 


CO 




+ 


+ 




»«• 


HM 




1 


1 

1 




1 


1 




C 8 






t- I- 






<M CN 





+ 

Mi 

CD 
I 

I 

4- 




4, ♦- j-J 4, O) 

02 CO H cc cw 



92 ELEMENTS OF ALGEBRA. 

Exercise 34. 

Find the cube roots of : 

1. x^— Sx^ + 5x^- 3x --1; x^- Saa^+ 5 a^ a^ 

— S a^ X — a^. 

2. 8 0^6 + 48 ax^+60 a^x'^-80 a^x^- 90 a^x^+lOSa^x 

- 27 a^. 

3. x^-6x^+ 15 x^-20 a^+15x^-6x+l. 

4. 27a^-54.a^h + 9a^^+2SaH^-3a^b^-6ah^-h^ 

5. Sx^+ 12 x^ -SOx*- 35 x^ + 4:5x^ + 27 x - 27, 

6. 216 + 3422:2+i7i^4 4.27^6_27^5_i09^-108ic. 

7. a3 - 3^25 - 53+ 8c3+ 6a2c-12a&c + 6fe2c 
4-12ac2- 125c2+ 3ah\ 

8. 1 - 3rK + 62^2 - 10 rt3 + 12 ic^ - 12 a;^ + 10 ri;^ - 6 a;^ 
4- 3 a:-8 - .^'9. 

9. 8 x^ - 36 x^y + 114 aj*?/^ _ 207 x^y^ + 285 0^23^ 
-225 0^2/5+125 7/6. 

10. a^ + 6a^h - Sa^c + 12 a V^ - 12 abc + 3ac2 
+ 8^3- 12h^c+ 6bc^-c^. 

11. x^ + Sx^y-Sa^y^-~lla^f+6xh/^+12xf - 8/. 

12. 204 rr*2/2 - 144 a;^ 7/ + 8 /- 36 o;^/^ _ 171 ^^s^^s _|. 54^6 
+ 102 0^2/*. 

43. Cube Root of Numerical Numbers. The method for 
extracting the cube root of arithmetical numbers is based 
upon the algebraic method. 



EVOLUTION. 93 

Since the cube root of 1000 is 10 ; of 1000000 is 100, etc., it fol- 
lows that the integral part of the cube root of numbers less than KXX) 
has one figure, of nuiubers between 1000 and 1000000 two figures, 
and so on. Hence, 

I/a point be placed over every third fgure in any number, beginning 
with units' place y the number of points will show the number of figures 
in the cube root. 

Thus, the cube root of 274625 has two figures ; the cube root of 
109215352 has three figures. 

If the given number contains decimals, the number of decimal 
places in the cube root will be one third as many as in the given 
number itself. Thus, if 1,11 be the cube root, the number will be 
1.367631 ; if .111 be the root, the number will be (3.00i36763i ; if 
11.111 be the root, the number will be 1371.706960631. Hence, 

The number of points to the left of the decimal point will show the 
number of integral places in the root, and the number of points to the 
right will show the number of decimal places. 

Example 1 . Find the cube root of 778688. 

a + 6 = 92 

Process. 778688 ( 90 + 2 -= 92 

The cube of a, or 90, 729000 

First remainder, 49688 

First trial divisor 3 a", or 3 (90)^ = 24300 

3 times the protluctof a and 6, or 3X90X2= 540 
Second term b of the root squared, 2*'* = 



First complete divisor, 24844 

First complete divisor X 6, or 2, 49688 

Explanation. There will be two ^gures in the root. Let a -f- 6 
denote the root, a being the value of the number in tens* place, and 
b the number in units' place. Then a must be the greatest nmltiple 
of 10 whose cube is less than 778688, this is 90.. Subtract a«, or 
the cu>)e of 90, from the given number. Dividing the remainder 
by 3 a*, or 24.3()0, gives 2 for the value of 6. Add to the trial divisor 
3 a 6, or 54(), and 6^, or 4, for the complete divisor. Multiplying by 
2 and subtracting, there is no remainder. Hence, 92 is the required 



94 



ELEMENTS OF ALGEBRA. 



root, because we have actually cubed it and subtracted this cube from 
the given number and found no remainder. 

Example 2. Find the cube root of 897.236011125. 



= 24300 

1620 

36 



25956 



Process. 

Cube of 9, 

First remainder, 

First trial divisor, 3 times (90) ^ 

3 times the product of 90 and 6, 

6 squared, 

First complete divisor, 

First complete divisor multiplied by 6, 

Second remainder. 

Second trial divisor, 3 times (960)2 - 2764800 

3 times the product of 960 and 4, 11520 

4 squared, 16 
Second complete divisor, 2776336 
Second complete divisor multiplied by 4, 
Third remainder, 

Third trial divisor, 3 times (9640)2 = 278788800 
3 times the product of 9640 and 5, 144600 

6 squared, 25 

Third complete divisor, 278933425 

Third complete divisor multiplied by 5, 



897.236011125(9.645 

729 

168236 



155736 
12500011 



11105344 
1394667125 



1394667125 



Let the student formulate a method for arithmetical cube root from 
what has been demonstrated. 

Note. The notes in Art. 41 are equally applicable to cube root, except that 
in Note 1 two ciphers must be annexed to the divisor instead of one. 



Exercise 35. 

Find the cube roots of: 

1. 74088; 34012.224; .244140626. 

2. ^mif^; .000152273304. 



EVOLUTION. 95 

Find to three places of decimals the cube roots of : 
3. .64; .08; 8.21; .3; .008; J; ^. 

44. Since a* = c^^ = (a*)* = Vo* = VVa, 

The fourth root is the square root of the square root. 

* 

Since a* = a^ '<3 = (aSf = Va* = V 4^, 

The sixth root is the cube root of the square root. Hence, 

When the root indices are composed of factors, the ope- 
ration is performed by successive extraction of simpler 
roots. 

Hote. It is suggested that the teacher use the remainder of this article at 
his discretion. 

We may find the fifth, seventh, eleventh, or any root of an 
expression or arithmetical number if desired, by using the 
form for completing the divisor. Thus, 

To find the fifth root. 

Form, (a + 6)» = a* + (5 a* + 10 a«6 + 10 a*^** -j- 5 a 6» + 6*) b. 

Trial divisor, 5 a*. 

Complete divisor, (5 a* -h 10 a» 6 + 10 a* &« + 5 a 6» + b*) . 

To find the seventh root. 

Form, (a+6)' = a7+(7a»+21a»H35a<6«-f35a»6»+21a26<+7a6»+6«)6. 

Trial divisor, 7 a*. 

Complete divisor, {7a*+2la^b+35a*b^+3ba%^-2la^b^-7ab^+h% 



96 



ELEMENTS OF ALGEBRA. 



Example. Find the fifth root of 36936242722357. 



Process. 




36936242722357(517 


a5 = 5S = 


3125 


First remainder, 


56862427 


First trial divisor =:5a'^(a considered 


as 




5 tens) = 5 (50)* = 


31250000 




10 a^b (h considered as 1 unit) = 






10(50)3 X 1 ::= 


1250000 




10a2 62= 10 X (50)-^X (1)^ = 


25000 




5 a 63 = 5X (50) X (l)^ ^= 


250 




6* = (1)^ 


1 




First complete divisor, 


32525251 


First complete divisor multiplied by 


1, 32525251 


Second remainder. 




2433717622357 



Second trial divisor = 5 a^ (a considered 

as 51 tens) = 5 X (510)^ = 338260050000 
10 a^b{b considered as 7 units) 

= 10 X (510)3 X (7) = 
lOa-^62 = 10 X (510)2 X (7)2 = 
5a68 = 5X(510) X (7)3 = 
6* = (7)4 

Second complete divisor, 347673946051 

Second complete divisor multiplied by 7, 2433717622357 



9285570000 

127449000 

874650 

2401 



Miscellaneous Exercise 36. 

Express the nth. roots of: 

1. ah^c-'^; 52-a;3"(^-;y + 2«)4"x2%^4-7/"y"x4"(a;— 2/T- 

2. Simplify 4: a (Sax y)^ — 5 xhj^ ^2S^ «^ X ak 
Find the square roots of: 

3. ^x + l^-\-x-^—4x-^^-l2x^. 

4. 28-24tt-^-16al + 9ft-" + 4a^ 



EVOLUTION. 97 

5. 162;'^"+16ic7--4a;8*-4x-9'' + a:io». 

6. a^u~l—4:xhj~^ + 6 — 4 x~^ yi + x~^t/^. 

7. 6«cr/;5 + 4 62u;* + a^x^^ -^ \) c^ -12 bcx^ - 4: abx'. 

8. {x'^ + Ax^ + ^ax^ -^ ia^-2x^-^ ax. 

9. a2»+ 2a\6"'+ z'""; « ± 2 ai a;i + x. 
Find the cube roots of : 

10. 60 x^i/ -^ 4S xf ^27 x^ + lOS x^i/ -90 si^y^ -{- 8/ 
-80a^^. 

11. 24a;*'" 2/2- 4- % 3^'^yin_ 6a^'»7/" + a:6m_9g^^n 
+ 64/"- 56a;3«2^* 

12. 15x-* - 6:r-i— 62;-6 + 15aj-2+ 1 + a:-6-^20a;-8. 

13. Su^-^2^f + ixi/-^f. 

14. ^a-i-6a-J-J + 8a-t-^a-8 + 27a-i + 54a-4 
+ |a-t + 36 a-4- 18a-2. 

Find the sixth roots of: 

15. 1215a*-1458a6+135a2-540a3_l8« + l+729a« 

16. .x^+ f-6x ?/+ 15jy^ij^-Gx^y+ 15 3^1/- 20 a^y^l 

17. 160 a3 + 240 a* + 60 a2 + 192 a^ + 64 a«+ 12 a + 1. 

18. 2985984; 262144. 
Find the eighth roots of : 

19. </8+28a3+8a+l + 56a8+70a* + 8aH56«6^28a«. 

20. («* + 5* - 2 a&8 + 3 a2i^ - 2 aH)*. 

7 



98 ELEMENTS OF ALGEBRA. -^ 

21. Find the 5tli root of 36936242722357. 

22. Find the 7th root of 1231171548132409344. 
Extract the following roots : 

23. y/(a4 + 19 a2 + 25 - 6 aS _ 30 a), 

24. y/[^^ — 2 {m + 71)0^ + (m^ + 4:mn -\- n^) x^ 

— 2mn (m -\- n) x + 7/^^ 71^] 2 

25. [25 a2 _ 20 a 5 + 4 52 + 9 c2 - 12 6c + 30 ac]i 

26. [27a^- 54a5+ 63 a^-Ua^^ 21a2_6a + l]i 

27. y/(aj2'" + 2 af"-^'^ - 2 ^'"^^ + ^2» _ 2 aj"+^ + ^). 

28. [a6_ 12^5+ 60^4- 160^3+ 240 a^-192a+ 64:]^. 

29. [(«4-fe)6'»^34.6^n^(^^j^4m^_^^2rt2V(a+&)2'"ic+8a3V]i 

30. y/(^2« ^ 2 a:2«-l + 3 3^2—2 _^ 2 a;2'»-3 + aj2n-4)^ 

31. y^(8 - 12a3«-i + 6a6"-2_a9«-3^^ 

Queries. What signs are given to even and odd roots ? Why ? 
What principles govern the signs of roots ? Upon what principle is 
the method lor finding the root of a monomial hased 1 How derive 
the method for finding the square root of any polynomial '? Why 
divide the first term of the remainder hy twice the terms of the root 
already found for the next term of the root ? Why add the quo- 
tient to the trial divisor for the complete divisor ? How derive the 
method for finding the cuhe root of any polynomial ? Why divide 
the first term of the remainder by three times the square of the root 
already found for the next term of the root ? Why add to the trial 
divisor three times the product of the terms of the root already found 
by the next term, and the square of the next term, for the complete 
divisor ? 



USE OF ALGEBRAIC S\MBULS. 99 



CHAPTER VIII. 
USE OF ALGEBRAIC SYMBOLS. 

45. Symbols of operation are used to indicate that 
algebraic operations are to be performed. 

Thus, m -f (a - 6) indicates that a — 6 is to be added to m ; 
m — (a — 6) indicates that a — h is to be subtracted from m. Per- 
forming the operations, we have, 

m-\- (a — b) = m + a — b ; 

m — (a — h) = m — a + b. Hence, 

A plus sign before a symbol of aggregation shows that the enclosed 
terms are to be added to what precedes ; as this operation does not 
change the signs, the removal of the symbol does not affect the signs. 
Removing one preceded by a minus sign changes the sign of each 
enclosed term. 



Thus, a-26-[4a-66-{3a-c+(5a-26-3a-c + 2 6){] 
= a-26-[4a-66-{3a-c + (5a-26-3a + c-2 6)}] 
= a-2 6-[4a-66-{3a-c+(2a-46 +c )}] 

= a-26-[4a-66-{3a-c+ 2a-46 +c |] 

= a-26-[4a-66-{6a -46 }] 

= a-26-[4a-66-5a +46 ] 

=a-26-[ -26- o ] 

= a-26 +26+ a 

= 2a 

Explanation. Remove the vinculum, subtract and unite like 
terras ; then remove the parenthesis and unite like terms ; now 
remove the brace, subtract and unite like terms ; finally, removing 
the bracket, subtracting and uniting like terms, we have 2 a. 



100 ELEMENTS OF ALGEBRA. 

Exercise 37. 

Simplify : 



1. 2a-l3b+(2b~c)-4:c+ {2 a-(Sh-c-2h)}']. 

2. a — b + c - (a + b — c) — (c — b- a). 

3. x^ ~[4x^~ {6 x^ - (4.r - 1)}] -(x^ + 4:x^+6x^ 
H- 4^ + 1). 

4. ~l0(x-\-y)-lz + x + y-3{x + 2y-(z+x-y)}^ 

4- 4:Z. 



5. a-[5b-{a-{5c-2c-b-4b) + 2a-{a-2b-hc)}l 

6. -5{a-6[a-{b- c)]} + 60 {6 - (c + a)}. 
7.-2a-(3b + 2c)-[5b-{6c-U) + 5c-{2a-(c+2b)}]. 



8. 3:r - {?/ - [?/ - (:r + ?y) - {- ;/ - (^^ _ ^ - ^z)}]}. 



9. 3a-[2b + a-b]-^ [Sb- 2a + bl 

10. {(re- 2 ?/ + a; ?/) - (a; - ?/ + ^)}-{a: -(x-2j + xy)}. 



11. I a - [| a - {1 « - (2 ft - 5 a + 6)} - (f a - 3)]. 

12. f{f(a-6)-8(&-c)}-{|(6-0-i(c-«)} 
- I {^ - a - |(« - 6)}. 

13. 5 {a - 2 [ft - 2 (ft + a;)]} - 4 {« - 2[a- 2 (a + rr)]}. 



14. ft+25-{6a~[3& + (8a;-2 + &.y-a; + 4a)]-3/;} 
+ 2(1 + |-«-46). 

15. 2 (|& - f ft) ~ 7 [ft - 6 {2 - 5 (ft - b}}\ 

16. -|{-f[-4(-:.i)]}+f{-3(-a;^)}. 

17. - I {- L" Q^ - h)]} + {- I [- (ft - fe)]}. 



USE OF ALGEBRAIC SYMBOLS. 101 

18. 5 {a - 2[6 - 3 (c H- d)]} - 4 {a - 3 [6 - 4 (c - </)]}. 

19. (a-l)(a - 2) - 3a(a + 3) + 2 {(a + 2)(a + 1)- 3}. 

20. {xz -{x'-y){y + z)} -y[y-{x- z)\ 

21. {a^h-\-c-\-df^{a-h-c^ df+ (a-b + c-d)^ 
+ (a + b-c- d)\ 

22. 7!'-{2xy-[-{x-{y-z}){x+{y-z})-\-2xy-\-4.yz} 

23. a(a + l)(aH-2)(a + 3)-6(2a-J)-"ta2-3a + l)2. 

24. 571 {(a:— y)a-&2;}— 27i{a:(a-6)-af ?/} - {Zax—{pzr-2x)h} n. 

25. (a:2 4.2^)^_(a; + y)(a;{7i-y}-y{^i-4). 

26. 2aVi — 3 m — [6yi — 6 7i + (2:i-2Vi)a] + &X Vy. 

27. (9 7?l2 7l2 - 4 71*) (7m2 - 7l2) - {3 77? 7?. - 2 W^} 
{3 771 (/?t2 4- 71^) — 2 71 (71* + 3 771 71 — TH?)} 71. 

28. 77l2 ( w2 + 7l2)2 - 2 7^2 ^2 (^ + n) (771 - 7^) - (tTI^ _ n3)2.. 

29. i(^^ + iy)(i^-Jy)-(J^-!.y)^-?(^-f2/^). 

30. Y a ^ + 4 ?/') ( J ^ - J ?/') - (i :r - 3) (J :r + 3) 
(4^-9) + (f y - 3)(f y + 3) (^ .7/2 - 9). 

The use of symbols of aggregation aid in shortening the 
work in certain cases in division. Thus, 

a 4- (6 + c) ) (6 + c) a^ + (62 -I- 6 c + c') a - - (6 + c) 6 c ( (6 + c) a - 6c 
(6 + c)a« + (62+ 2 6c + c«) a 



+ ( - 6c )a-(6 + c)6 
+ ( - 6c )a-(6 + c)6 



102 ELEMENTS OF ALGEBRA. 

Divide : 

31. (6 + c)a2 + (&2 + 32^c + c2)a+(i + c)&c by a + h + c. 

32. (a -\-hf-6{a + b)- 27 by (a + b) + 3. 

33. {x+ijf+ 3 {x + yfz + S{x + y)z^+^ by (x + yf + 
2(x + y)z + z^. 

34. (x + yf _. 2 (^ + ?/) z + z^ by x ^ y — z. 

35. {a + &)3 + 1 by « + J + 1. 

46. The converse operation of enclosing any number 
of terms of an expression in a symbol of aggregation is 
important. 

a + m~c-\-h — n = a + m — c-\-(h — 71). 
a — m — c -]- b ~ n = a — (m + c) -^ (b — n). 
ax^ — ny + hx^ — cy^= {ax^ — ny) + (hx^ — c y*). 
xy — ax'-by + ab = (x y — by) — (a x — a b). 

Hence, when the signs + and — indicate operation : 

(1) Any number of terms may be enclosed in a symbol of aggrega- 
tion preceded by the sign + , without changing the sign of each term. 

(2) Any number of terms may be enclosed in a symbol of aggrega- 
tion preceded by the sign — , if the sign of each term be changed. 

The terms may "be enclosed in various ways. Thus, 

am + an — ax — bx + cy — dz=(am — ax) + \an — bx^-\-\cy — dz\, 
or, am-\-an—ax — bx-\-cy—dz= (am, + an — ax)—{bx — cy + dz], 
or, am + an — ax — bx+cy — dz= {am-\-an)—{ax-\-bx)+{cy—dz). Etc. 

If a factor is common to each term within a symbol of aggregation, 
it may be placed outside as a multiplier. Thus, 
ax^ + bx'^~bx^ + dx'^={ax^--bx^) + {bx'^+dx^)=x^{a-b)+x'^{b + d). 

Note. An expression consisting of three or more terms may be raised toa 
given power by inspection, by first changing it to the form of a binomial. 
Thus, (a + & + c — (^)4 = [(a + J) + (c — d)Y^ = etc. 



USE OF ALGEBRAIC SYMBOLS. 103 

Exercise 38. 

Bracket the last three terms so that each bracket shall 
be preceded by a — si<,'n : 

I. a:* - a x8 - 5 a^» + 2 ; m6 + 3 w3 + 3 - 6 m2. 

3. 4:X+'dao(^—{)7^ — bey -^ y; x^—y^—z^-\-ab-\-3ac. 

4. Express each of the above as binomials, and enclose 
the last two terms in an inner brace preceded by a — sign. 

Bracket the following in binomials, also in trinomials, 
each preceded by a — sign : 

5. 2ab — Say-\-4:bz — 5bx-2cd — 3. 

6. a— 26 + cz — d — l-^z — x—2y+ 2m — n-\-p—4ahc. 

7. 2x-Sxy-{- 4 ^f - 5 2:3^2 + ^,p _ ^.y 2 

8. a^+3 a*— 4 a^—3 CT^-f a —1 ; —2 m—2> 7i+4p— 5 2;— 1—6 y. 

9. a n ■\- ab — a c — c X — a X — a y — 3 ab c -\- Z xy z. 

10. Express the above six examples in trinomials, and 
enclose the last two terms in an inner bracket preceded by 
a — sign. 

II. Expand (tw + 2 71 — j^. 

12. Simplify and bracket like powers of x in 2b^—ax 
-{a:i^-\bx-nx- {a^ -^ 3r.i2}-| _ {ax'-2cx)}. 

Queries. Why may a 8ymlx)l of aggregation preceded by a + 
sigu be remove<l without changing the signs of the enclosed terms ? 
If a symbol of aggregation pi*ecedetl by a — sign be removed, why 
change the signs of the enclosed terms? 



104 ELEMENTS OF ALGEBRA. 

CHAPTEE IX. 

SIMPLE EQUATIONS. 

47. 3 a: + 5 = 5 a; — 7 is called an Equation. The first 
memher or first side is 3^ + 5, and the second member or 
second side is 5 x — 7. 

x = x, 14 = 14, are called Identities or Identical Equations. 

To solve an equation is to find the value of the unknown 
number. 

The process of solving an equation depends upon the 
following axioms: 

1. If to equal numhers we add equal numbers, the sums 
are equal. 

2. If from equal numhers we subtract equal numbers, the 
remainders are equal. 

3. If equal numbers are multiplied by equal numbers, the 
products are equal. 

4. If equal numhers are divided by equal numbers^ the 
quotients are equal. 

Example 1 . Find the value of x in the equation 6 a:— 11 — Sx+lO. 

Solution. Subtracting 3 x from each member of the equation 
(Axiom 2), we have 6a; — 3a;— ll = 3cc — 3.X+10. Uniting like 
terms, 3 a; — 11 = 10. Adding 11 to each member (Axiom 1), and 
uniting like terms, we get 3 a; = 21. Dividing both members by 3 
(Axiom 4) gives x—1. 

Proof. To verify this result, substitute 7 for x in the given equa- 
tion. Then, 6 X 7 - 11 = 3 X 7 + 10, or 31 = 31, which is an 
identity. Hence, the value of x is 7. 



SIMPLE EQUATIONS. 105 

Example 2. Solve the equation 2 (a: - 8) - 3 (9 - x) + 5 (a: - 1 1) 
= 7 -3 (a: -17). 

Solution. Performing the indicated operations, and uniting like 
terms, 10 a; — 98 = 58 — 3 z. Adding 3 x and 98 to each member of 
the equation, we have 10 « + 3 a: - 98 + 98 = 58 + 98 - 3 a: + 3 x, 
or uniting like terms, 13 x= 156. Dividing both members by 13, 
X- 12. 

Proof. Substitute 12 for x in the given equation. 

Then, 2(12-«) - 3 (9 - 12^ + 5 (12 - 11) = 7 - 3 (12 - 17), 
or, 8 + 9 + 5 = 7 + 15, 

or, 22 = 22, an identity. 

Therefore, the value of x is 12. 

Example 3. Solve the equation 14 — x — 5 (x — 3) (x -f 2) 
+ (5 - x) (4 - 5 x) = 45 X - 76. 

Process. Simplify, 64 - 25 x = 45 x - 76. 

Subtract 45 x, 64 - 70 x = - 76. 

Subtract 64, - 70 x = - 140. 

Divide by — 70, x = 2. 

Notet : 1. To verify, that is, to jncne the truth of the result^ substitute the 
supposed value of the unknown number in the given equation and thus find if it 
satisfies its conditions. 

2. In simplifying an equation the student should be careful to notice that 
when the sign — precedes a term, in removing the symbol of rggregation, the 
sign of each term must be changed. 

Exercise 39. 
Solve the following equation.s : 
1. 6a;+ 1 = 5a:-f 10; 11 - 7z = 18a:- 14. 

3. 2r+3= ir,-(2ar-3); 3 (a:- 2) + 4 = 4 (3 - a:). 

4. 7(a:- 18) = 3(x- 14); 7a:+6-3a: = 56 + 2.r. 

5. 15 (x - 1) + 4 (x + 3) = 2{x+ 7). 

6. 5 - 3 (4 - a:) + 4 (3 - 2 x) = 0. 



106 ELEMENTS OF ALGEBRA. 

48. If we add the same number to each member of an 
equation, or subtract it from each member, the results are 
equal, each to each. Thus, 

Consider the equation x — b = a. Adding b to each side, we get, 

X = a -{- b. 
Consider the equation x + b = a. Subtracting 6 from each side, 
we have x = a — b. 

In each case b is transposed from one side to the other, 
but its sign is changed. Hence, 

Any term may he transposed from one side of an equation 
to the other, provided its sign be changed. 

Example. Solve (a: + 1) (x + 2) (x + Q) - (x - 2) (x -\- 2) 
= x^ + 9x^ + 4(7x-l)-{- (2-x)(3 + x). 

Process. Simphfy, x^ + 8x^ -}-20x-\-\6 = x» + 8x^-}-27x + 2. 
Transpose, x^ - x^ + 8x^ ~ 8x^ + 20 x-27 x = 2 - 16. 
Unite Hke terms, — 7 x = — 14. 

Divide by — 7, x = 2. 

* Hence, in general, 

To Solve a Simple Equation of one Unknown Number. If 

necessary, simplify the equation. Transpose all the terms 
containing the unknown number to one side, and all other 
terms to the other side. Unite like terms, and. divide both 
sides by the coefficient of the unknown number. 

Exercise 40. 

Solve the following equations : 

1. 12 X - 20 a: + 13 - 9 2^ - 259 ; 336 + (3 :c - 1 1) 

= 2 (5 2: - 5) + 8 (97 - 7 a:). 

2. 62; + 4a; = 3:r + 84; 6a: + 2(13-2^) = 3 (17 -a;). 



SIMPLE EQUATIONS. 107 

3. 2 (a; +2) + 182: = 3(5 + 2;) + 0; 30 a; + 20 2:- 15 a; 
+ 12 a; = 2820. 

4 9(2; _ 1) + 2 (a: -2) =10 (2 -a:); 2 (a: + 2) (a: - 4) 
= a;(2a;+ 1)-2L 

5. 6y-2(9-4?/) + 3(52/-7) = 10 2/-(4 + 16y+35), 
and verify. 

6. 2y-(4y-l) = 5y-(//+l); 56+ 21 a;- 8 (2 a;- 1) 
= 62. 

7. 10 [224 -{x-V 192)] = 7 (28 + 3a:); 9 (7 + ^y) 
-4[9-(2-.y)] = 252y. 

8 25 a:- V.) - [3-{4a:-3}] = a: — (a: — 5), and verify. 

9 20(2~a;) + 3(a:-7)-2[a;+9-3{9-4(2-a:)}] = 1. 

10. (y-2)(7-7/) + (7/-5)(y+3)-2(y^l) + 12 = 0. 

11. 4 (v/ + 5)2 - (2 y + 1)2 = 3 (y - 5) + 180 ; 2.25 x 
- 1.25 = 3 a: + 3.75. 

12. .15?/ + 1.575 -.875?/ = .0625 2^. 
Query. In transposing, why change the signs ? 

49. Known Numbers are represented by the first letters 
of the alphabet, and by figures ; as, a, b, 2 c, 6. 

Unknown Numbers are usually represented by the last 
letters of tlie alpliabet; as, x, y, z. 

An Equation i.s a statement that two expressions repre- 
sent the same number. 

An Identical Equation, or an Identity, i.s one which is 
true for all values of the letters which enter into it ; as, 
{a + z) (a — a:) = a2 — 3^. 



108 ELEMENTS OF ALGEBRA. 

The Roots of an equation are the values of the unknown 
numbers. 

The Degree of an equation is the power of the unknown 
number, and is determined by the greatest number of un- 
known factors in any term. 

Thus, X — y = 6 is an equation of the Jirst degree ; 4x^ + 5y = 3 
and 5 xy + 2 — 3 X ave equations of the second degree. 

A Simple Equation is an equation of the first degree. 

Miscellaneous Exercise 41. 

Solve the following equations : 

1. 5(7 + 3:?/)-(23/-3)(l-2?/)-(2,y-3)2 + (5 + 2/) - 0. 

2. {22j+lf + {27/-lf=UyQf-4:)+57, and verify. 

3. 1.5(26i/-51)-12{l-3y)=7Sy-2[5y-2.6{l-.Sy)]. 

4. .6a:-.7:r + .752:-.875:r + 15 = 0; .6y-{.lSy-M) 
= .2 2/ + 4.45. 

5. 30 ^ - 3 [30 z-{2z- 5)] = 5{2z- 57) - 50. 

6. 10 (^ + 10) - 18 (3 2 - 4) + 5 (3^-2) (2 z - 3) 
= 30 2;^ — 16, and verify. 

7. 4.Sy-2{.72y-M) = 1.6?/ + 8.9; .5x-.3x-.25 
= .25a:-l. 

8. .2 7/- .16?/ = .6 -.3; .5y - .2y = .3y- 15. 

9. 5.6 y f .25 y+ .3y = y-'S; .6 y -\- .25 - A y = l.S 
-.75?/ -.3. 

10. 3)x- .25 (x-2)~- .3 (3x-{- 12)? = 41. 



PROBLEMS LEADING TO SIMPLE EQUATIONS. 109 



CHAPTER X. 
PROBLEMS LEADING TO SIMPLE EQUATIONS. 

50. The beginner will find the model solutions of great benefit 
ill forming statements, and he should give them careful consideration 
before attempting to solve any of the problems in each set. 

Exercise 42. 

1. A father is 35 and his sod 8 years old. In how 
many years will the father be just twice as old as the 
son ? 

Solution. Let x — the number of years required. 
Then x + 35 = the number of years in the father's age x years 

from now, 
and X -f 8 = the number of years in the son's age x years 

from now. 
By the conditions of the problem, at the expiration of x years 
twice the son's age, or 2 (x + 8), equals the father's age, or a: -f 35. 
Hence, the equation 2 (x + 8) = x + 35, or 2 x + 16 = x -I- 35. 
Transposing and uniting like terfns, x = 19. 

2. One number exceeds another by 5, and their sum is 
29. Find the numbers. 

3. The difference of two numbers is 14, and their sum 
is 48. Find the numbers. 

4. A father gave S200 to his five sons, which they are 
to divide according to their ages, so that each elder son 
shall receive $10 more than his next younger brother. 
Find the share of each. 



110 ELEMENTS OF ALGEBRA. 

5. A father is four times as old as his son ; in 24 years 
he will only be twice as old. Find their ages. 



6. Divide 50 into two parts, so that three times the 
greater may exceed 100 by as much as 8 times the less 
falls short of 120. 

Solution. Let x = the greater part. 

Then 50 — x = the less part, 

and 3 X = three times the greater part ; 

also, 8 (50 - x) = eight times the less part. 

But, 3 X — 100 = the excess of three times the greater part over 
100; 
also, 120—8 (50 — x) = the number that eight times the less lacks of 
120. 

By the conditions, 3 a: - 100 = 120 - 8 (50 - x). 

Therefore, . x = 36, for the greater part, 

and 50 — a; = 14, for the less part. 

7. Twenty-three times a certain number is as much 
above 14 as 16 is above seven times the number. Find 
the number. 

8. A is five years older than B. In 15 years the sum 
of their ages will be three times the present age of A. 
Find the age of each. 

9. A is 25 years older than B, and A's age is as much 
above 20 as B's is below 85. Find their ages. 

10. The sum of the ages of A and B is 30 years, and 
five years hence A will be three times as old as B. Find 
their ages. 

11. The difference between the squares of two consecu- 
tive numbers is 121. Find the numbers. 



PROBLExMS LEADUSG TO SIMPLE EQUATIONS. Ill 

Solution. Let x = the less number. 
Then will jc -f 1 = the greater number, 

x^ = the stjuare of the less number, 
and (x -I- 1)*^ = the stpuire of the greater number. 

Then (x + 1)^ - x'^ = the ditlerence of the scjuai-e numbers. 
But 121 = the difference of the squares. 

Hence, (x + l)^ - x- = 121. 
Therefore, x = 60, the less number, 

X -h 1 = 61, the greater number. 

12. Find three consecutive numbers whose sum is 27. 

13. The difference of two numbers is 3, and the differ- 
ence of their squares is 21. Find the numbers. 

14. Find a number such that if 5, 15, and 35 be added 
to it, the product of the first and third results may be equal 
to the square of the second. 

15. I sold a cow for S35 and half as much as I gave for 
it, and gained SIO. Find the cost of the cow. 



16. A had four times as much money as B ; but, after, 
giving B $16, he had only two times as much as B. How 
much had each at first ? 

Solution. Let x = the number of dollars that B had at first. 
Then 4x = the number of dollars that A had at first. 

But 4x — 16 = the number of dollars that A had after giving 

B^16, 
and X + 16 = the number of dollars B had after receiving 

^16 from A. 
By the conditions, 4 X- 16 = 2 (x + 16). 

Therefore, x = 24, the number of dollars that B had, 

and 4x = 96, the number of dollars that A had. 

17. A father is 3 times as old as his son ; four years 
ago the father was 4 times as old as bis son then was. 
Find their ages. 



112 elp:ments of algebra. 

18. One number is two times another; but if 50 be 
subtracted from each, one will be three times the other. 
Find the numbers. 

19. A has $26.20 and B has $35.80. B gave A a cer- 
tain sum; then A had four times as much as B. How 
much did A receive from B ? 

20. If 288 be added to a certain number, the result will 
be equal to three times the excess of the number over 12. 
Find the number. 

21. A farmer has grain worth $0.60 per bushel, and 
other grain worth $1.10 per bushel. How many bushels 
of each kind must be taken to make a mixture of 40 
bushels worth $0.90 a bushel? 

Solution. 

Let X = the number of bushels required of the f 0.60 grain. 

Then 40 — a: = the number of bushels required of the $1.10 grain; 

and Y®^*\j X — the number of dollars in the cost of the $0.60 grain; 

also, 1.10(40-a:) — the number of dollars in the cost of the $1.10grain. 

Hence, ^-^^ a; -f- 1.10 (40 — x) — the number of dollars in the total 

cost of the mixture. 
But the cost of the mixture is to be $36. Hence, 

^%x-\- 1.10 (40 -a:) = 36. 
Therefore, x = 16, the number of bushels of the $0.60 kind, 
and 40 - X = 24, the number of bushels of the $1.10 kind. 

22. A merchant has two kinds of vinegar : one worth 
$0.35 a quart and the other $1.25 a gallon. From these 
he made a mixture of 63 gallons, worth $1.30 a gallon. 
How many gallons did he take of each kind ? 

23. A merchant has a mixture of 88 pounds of 13 and 
11 cent sugar, which he sells at 12| cents per pound. 
How many pounds of each kind are there ? 



PROBLEMS LEADING TO SIMPLE EQUATIONS. 113 

24. I bought 24 pounds of tea of two different kinds, 
and paid for the whole $9. The better kind cost $0.65 
per pound, and the poorer kind $0.35 per pound. How 
many pounds were there of each kind ? 

25. A grocer having 75 pounds of tea worth $0.90 a 
pound, mixed with it so much tea at $0.50 a pound that 
the combined mixture was worth $0.80 a pound. How 
much did he add ? 

Remarks. No general method can be given for the solution of 
problems. ^ 

The beginner will find that his principal difficulty in solving a 
problem consists in forming the equation of conditions, and in order 
to overcome this, much will depend upon his skill and ingenuity. 

The statement of a problem consists in translating its conditions 
into algebraic symbols and ordinary language. Many times the be- 
ginner fails to form a correct statement, because he does not under- 
stand what is meant by the ordinary language of the problem. If he 
cannot assign a consistent meaning to the words, it will be impossi- 
ble for him to express their meaning in algebraic symbols. It often 
happens that the words appear to be susceptible of more than one 
meaning. In such caj^es the student should express the meaning that 
seems most reasonable in algebraic symbols, and obtain the result to 
which it will lead. Should such result be inadmissible, the student 
should Uy another meaning of the words. 

The student must depend upon hus own powers^ and should he at 
times be perplexed, he must not be discouraged, since nothing but 
patience and practice can overcome the difficulties and give him 
readiness and certainty in solving problems. He must study the 
iT»eaning of the language of the problem, to ascertain the unknown 
numbers in it. There may be several such numl^rs, but oftentimes a 
little skilful manipulation will enable one to express all of the un- 
known numbers in terms of some one of them. Select the one by 
which this can be most easily done and represent it by some one of 
the final letters of the alphabet. 

Among the following problems no doubt the beginner will find 

8 



114 ELEMENTS OF ALGEBRA. 

some which he can readily solve by arithmetic, or by guessing and 
trial; he may thus be led to undervalue the power of algebra, and to 
regard its aid as unnecessary. In reply, as the student advances lie 
will find that by the aid of algebi'a he can solve not only all of these 
problems, without any uncertainty or guessing, but those which 
would be exceedingly difficult, if not altogether impossible, if he 
depended upon arithmetical processes alone. 

26. A's age is six times B's, and fifteen years hence A 
will be three times as old as B. Find their ages. 

27. A is three times as old as B, and 12 years since he 
was fiv^ times as old. Find B's age. 

28. A father has three sons; his age is 60, and the 
joint ages of the sons is 46. How long will it be before 
the joint ages of the sons will be equal to that of the 
fathei' ? 

29. If yon walk 10 miles, then travel a certain distance 
by train, and then twice as far by coach, and the whole 
journey is 70 miles, how far will you travel by coach? 

30. A is twice as old as B, and seven years ago their 
united ages amounted to as many years as now represent 
the age of A. Find their ages. 

31. After 136 quarts had been drawn out of one of two 
equal casks, and 80 gallons out of the other, there remained 
just three times as much in one cask as in the other. 
Find the contents of each cask. 

32. Find the number whose double increased by 1.2 
exceeds 3.65 by as much as the number itself is less than 
8.65. 

33. Find three consecutive numbers such that if they 
be diminished by 10, 17, and 26, respectively, their sum 
will be 10. 



PROBLEMS LEADING TO SIMPLE EQUATIONS. 115 

34. Two consecutive numbers are such that one fourth 
of the less exceeds one tifth of the greater by 1. Find the 
numbers. 

35. There are two consecutive numbers such that one 
fifth of the greater exceeds one tenth of the less by 3. 
Find tliem. 

36. Find a number such that the sum of its half and 
its fourth shall exceed the sum of its fifth and its tenth by 
45. 

37. Find a number such that the sum of its half and 
its fifth shall exceed the difiference of its fourth and its 
tenth by 110. 

38. If a watch and chain are worth $185, and the 
watch lacks $19 of being worth two times the cost of the 
chain, find the cost of each. 

39. If silk costs 6 times as much as linen, and I buy 

22 yards of silk and 28 yards of linen at a cost of $52, 
find the cost of each per yard. 

40. A man gave 17 boys $3.31, giving to some 13 cents 
each and to the rest 23 cents each. How many received 

23 cents ? 

41. I paid a bill of $1.53 with 39 pieces of money, 
some 3-cent and the rest 5-cent pieces. How many of 
each did it take ? 

42. A son earns 37 cents per day less than his father, 
and in 8 days the father earns $6.08 more than the son 
earns in 5 days. Find the daily wages of each. 

43. How many 10-cent pieces and how many 25-cent 
pieces must be taken so that 95 pieces shall make $12.35? 



116 ELEMENTS OF ALGEBRA. 

44. Divide $112 into two parts, so that the number of 
five-cent pieces in one may equal the number of three-cent 
pieces in the other. 

45. A sum of money consists of dollars, twenty-five- cent 
pieces, and dimes, and amounts to $29.50. The number 
of coins is 55. There are twice as many dimes as quarters. 
How many are there of each kind ? 

46. A sum of £8 17 s. is made up of 124 coins, consist- 
ing of florins and shillings. How many are there of each? 

47. A bill of £4 5s. was paid in crowns, half-crowns, 
and shillings. The number of half-crowns used was four 
times the number of crowns and twice the number of shil- 
lings. How many were there of each ? 

48. A bill of £48 J was paid with guineas and half- 
crowns, and 12 more half-crowns than guineas were used. 
How many were there of each ? 

49. A company of 84 persons consists of men, women, 
and children. There are three times as many men as 
women, and five times as many women as children. How 
many are tliere of each ? 

50. The sum of three numbers is 263. The first is 3 
times the second, and the third is 23 more than 5 times 
the sum of the other two. Find the numbers. 

51. A farmer wishes to mix 660 bushels of feed, con- 
taining oats, corn, rye, and barley, so that the mixture 
may contain two times as much corn as oats, three times 
as much rye as corn, and four times as much barley as 
rye. How many bushels of each should be used ? 



PROBLEMS LEADING TO SIMPLE EQUATIONS. 117 

52. Divide $2590 into two sucli parts that the first at 
7% simple interest for 8 years may amount to the same 
sum as the second in 5 years at 8 %. 

Note. The character % is sometimes used for the term "/>er cent." Per 
cent is used by ellipsis for rate per cent. Thus, au allowauce of 7 on a hundred 
is at a rate of .07, aud the rate per cent is 7. 

53. $330 is invested in two parts, on one of which 
15% is gained, and on the other 8 % is lost. The total 
amount returned from the investment is S345. Find the 
investment. 

54. A man has $ 7585. He built a house, aud put tho 
rest out at simple interest for 18 months; 40% of it at 
5 % and the remainder at 6 %. The income from both in- 
vestments is $211.26. Find the cost of the house. 

55. In a certain weight of gunpowder the saltpetre was 
4 pounds less than half the weight, the sulphur 5 pounds 
more than a fifth, and the charcoal 3 pounds more than 
a tenth. Find the number of pounds of each. 

56. A company of 266 persons consists of men, women, 
aud children. Tlie men are 14 more in number than the 
women ; the children 34 more than the men and women 
together. How many are there of each ? 

57. I bought 16 yards of cloth, and if I had bought one 
yard less for the same money, each yard would have cost 
$0.25 more. Find the cost per yard of the cloth. 

68. A and B, 85 miles apart, set out at the same time 
to meet each other; A travels 5 miles an hour aud B 4 
miles an hour. How far will each have travelled when 
they meet ? 



118 ELEMENTS OF ALGEBRA. 

59. $330 is loaned for nine months in two parts ; on 
one 15 % per annum is gained, and on the other 8 % 
per annum is lost. The total amount from the loan is 
$364.25. Find the amount in each loan. 

60. A boy has a certain sum of money, lie borrowed as 
much more, and spent 12 cents; he again borrowed as 
much as he had left, and spent 12 cents ; again he bor- 
rowed* as much as he had left, and spent 12 cents ; after 
which he had nothing left. How much money had he at 
first? 

61. A carriage, horse, and harness are worth $720. The 
carriage is worth eight tenths of the value of the horse, and 
the harness six tenths of the difference between the value 
of the horse and carriage. Find the value of each. 

62. A boy sold half an apple more than half his apples. 
Again he sold half an apple more than half his remaining 
apples. A third time he repeated the process; and he had 
sold all his apples. How many apples had he ? 

Algebra is the science which treats of algebraic liumbers 
and the symbols of relation. 

Algebra, like arithmetic, is a science which treats of numbers. In 
arithmetic the numbers are positive and represented by figures. In 
algebra the letters of the alphabet or figures are used to represent 
numbers, and they may be positive or negative, real or imaginary. 

Algebra enables us to prove general theorems respecting numbers, 
and also to express those theorems briefly. 



FACTORING. 119 

CHAPTER XL 
FACTORING. 

51. A Factor is one of the makers of a number. 

Thus, since 5 with the aid of 4 and by the process of multiplica- 
tion makes 20, 5 is a factor of 20. 

A factor is also a divisor, but it is considered a divisor when it 
separates a number into parts, not when it helps to make up a 
number. 

Note. Unity cannot be a factor. 

Factoring is the process of separating an expression into 
its factors. 

Example. Find the factors of 12 a« 6 x^. 

Solution. The prime factors of 12 are 2, 2, and 3. The factors 
of a' are a, a, and a. The factors of x^ are x and a:*. 

Therefore, l2a*bxi = 2X2X3xaXaXaXbXxXxK 
Hence, a» a direct result of the principle that monomials are mul- 
tiplied by writing the several letters in connection, and giving each 
an exponent equal to the sum of the exponents of that letter in the 
factors. 

To Factor a MonomiaL Separate the letters into any number 
of factors, so that the sum of all the exponents of each factor shall 
make the exponent of that factor in the given expression ; also sepa- 
mte the numerical coefficient into its prime factors. 

Exercise 43. 

Separate into factors with integral exponents : 

1. Ua^l^x; Wo^t/^; 15ah^(^; 20ab(^; Soa^f:!^^; 



120 ELEMENTS OF ALGEBRA. 

Separate into two equal factors : 

2. UaH'^; da^f; SI a^ b^ x^"" y^"" ; 169a"5. 
Eemove the factor 2 a^ bi from : 

3. Sa^b; Qabx; IQab^c^; 10 a'H-^ x^y^. 
Separate into three factors, also into four : 

4. cc ; m^ " ; a" ; xi ; x^. 

52. Example 1. Factor a^x - Sa^x^. 

Solution. Dividing the expression by a^ x, we have a — 3 aj. 
Hence, a^ x — 3a^x = a'^x {a — 3 x). 

Example 2. Factor 5 a'^b^x^ - 15 ab^x^ + 20 b« x^. 

Solution. By examining the terms of the expression we find 
that 5 b^ x^ is a factor of every term. Dividing by this common fac- 
tor the other is fomid. Hence, the factors are 5 h^x^ and a^x — Sax 
+ 4 6. 

.-. 5aH^x^-l5ab^x»-\-20b^x^ = 5b^x^(a^x-3ax + 4b). 
Hence, 

When the Terms of a Polynomial have a Monomial Factor. 

Divide each term of the expression by the common factor. The 
divisor and quotient will be the required factors. 

Exercise 44. 

Factor the following : 

1. 7n^ + n; 4: a^b + ab^c+ Sab; Sa^- 12 a^. 

2. ax — bx+cx;S9a^7/-{-57x^y^. 

3. 5x^ + Sa^-x^; 72 h^ x'^y^ - 84:b^ x y^ - 9Q a b a^y^. 

4. 924 «2 x^y^'z- 1 178 a x"" y z"" + 1232 a^ x"" y'^ z\ 



FACTORING. 121 

5. 4:aH-(J0aI^+20abc-{-SaH*x^+Uahy-Z6aHcx^, 

6. 2 xi y ^ a b X y + c a^ y^ ; 5 x^ + 10 x^ — 15 xi. 

53. In certain Trinomiala, of the form x^ -\- ax -^ b, where a 
and b represent any numbers, either integral, fractional, positive, or 
negative, it is possible to reverse the operation of Art. 25, and sepa- 
rate the expression into the product of two binomial factors. Evi- 
dently the first term of each factor will be the square root of a;*, or x; 
and to obtain the second terms of the factors, /ind two numbers whose 
algebraic product is the last term, or b, and whose sum is the coefficient 
ofx, or a. 

Example 1. Factor x^ + 21 a: -{- 110. 

Solution. Evidently the first term of each factor will be x. The 
second term of the factors must be two numlnirs whose product is 
110 (the third term), and whose sum in 21 (the coefficient of a;). The 
only two numbers whose product is 110 and whose sum is 21 are 10 
and 11. Therefore, z« + 21 1+ 1 10 = (a: + 10) (x + U). 

Example 2. Factor x^ + x- 132. 

Solution. Evidently the first term of each binomial factor will 
be X. The second term of the two binomiul factors must be two 
numbers whose algebraic product is — 132 and whose sum is -f- 1 
(the coefficient of x). The only two numbers whose product is — 132 
and whose sum is -f 1 are 4- 12 and — 11. Therefore, x^ -\- x — 132 
= (X -\- U) (x - U). 

Example 3. Factor y^ - 5cy — 50 c^ 

Solution. Evidently the first terra of each binomial factor will 
be y. The second term of the two binomial factors must Ijc two 
numbers whose product is — 50 c* and whose sum is —5c (the coef- 
ficient of y). The only two numbers whose pro<luct is — 50 c* and 
whose sum is —5c are + 5 c and — 10 c. .-. y^ — 5 cy — 50 c* 
= (y-l-5c)(y-10c). 



122 ELEMENTS OF ALGEBRA. 

Example 4. Factor x^y^ — (jn — n) xij — m n. 

Solution. Evidently the first term of each binomial factor will 
ho, xy.. The second term of the two binomial factors must be two 
such numbers whose product is —nin and whose sum is — (in — n). 
The only two numbers whose product is — m n and whose sum is 
— (m — n) are + n and — m. 

.*. x^y'^ — (in — n) xy — m n = (x y -\- n) {xy — m). Hence, 

I. If the Coefficient of the Highest Power is Unity. For the 

first term of each factor take the square root of one term of the trino- 
mial ; and for the second term of the factors, such numbers that 
their algebraic product will be another term of the trinomial, and 
their sum multiplied by the first term of either factor will be the 
remaining term of the trinomial. 

Exercise 45. 

Factor the following : 

1. ^24.19^+88; .^2-72^+ 12; ft8_20a4 + 96. 

2. 2:2 + 35 a; + 216; 52(^2- 245c + 143. 

3. ft* ¥ + 37 a2 62 + 300 ; a^ + 5 ah - 66 h\ 

4. a?ly^-oah-24.- ft* + 15*2 + 44. a^ + 17 ft^ + 60. 

5. a^y---dahc-lQc^; a^-2a'^-120; yi2+.8 ^i + l.o. 

7. ^2 _ 15 a; + 44; ^„2 + .i_i. ,„, ^ i^_ . ^2 _ 11 ^^ _ 26. 

8. 130 + 31 ft & + ft2 /;2 . ,,2 - 20 ah x+ 75 h'^ oj^ ; y^ 
+ 6 a;2 7/2 _ 27 2)4; 1 + 13 ^ + 42 :r2 ; ^;i2 - 15 a m + 06 a^. 

9. ft2 - lSax7j - 243 x^ y^; (x + yf + 5 {x +. ?/) + 4. 

10. 40 ft2 2,2 _ 13 ,-^ 7, + 1 . (^ _ 5)2 + (^ -h)-2. 



FACTORING. 123 

11. (x-yf-d{x-y)-l0;ay^+54x-]'729', 204-29^^+ .,4. 

12. (a -hiy -^9 (a + i)2 + 8 ; 2;*" - (6 +'m) x"^"" -^ b vi. 

13. a2 - 10 a V^v - 39 h^c^- x^- -^ {a - h) x"" - a b. 

14. a^-9xij-70f', a;2-.|.c-|; 2;*''-43ic2n^46Q 

15. 2^-{-iax--^^a^; x*-a^x'^-4e2a\ 

16. x^i/-^Sx7/-154:; a^" x*"' + 14 a" x^"' 2/" + 33 y^. 

17. a.>2 ?/2 _ 28 a" i" a;?/ + 187 a^- b^- - x^^\x + '^^V- 

18. a;*"//" + 20a'"^/'";;t2''7/2n ^ 5irt2«.j2m. (^ ^ ^^^em 
-- 7 a*" (a; + ?/)3- - 98 a^" ; n* + .01 n^ - .011. 

19. ^+^x-^\; u;2+2^2/~.21y2; «4+_8^^2+ j^. 

By an extension of the foregoing principles we may factor some 
trinoniials, of the form c^x^ + ax -\- hd^ where the coefficient of a:^ is 
a perfect square. Thus, 

Example 20. Factor 4 a:« + 4 a: -^ 3. 

Solution. The first term of each binomial factor will be the 
square root of 4 x^. The second term of the two binomial factors 
must >>e two numbers whose product is — .3 and whose sum multiplied 
by 2 a; is + 4 x. The only two numbers whose product is — 3 and 
whose sum multiplied by 2 a: is + 4 a; are + 3 and — 1 . 

.-. 4a:« + 4a;-3=(2x + .3)(2a:-l). 

21. 4a.^-10:r + 6; 9a^»-27a;+18; 4a'2+16aa;+12a2 

22. 9 rt2 + 30 « 5 + 24 J2; 16 a^« - 20 a a; + 6 «« 

23. 25.xiO'"-|2r5"'rt''-Ja2-; 36(a-t)*- + 12(a-6)*" + * 
- 143 (a - h)\ 



124 ELEMENTS OP ALGEBRA. 

54. We may factor some trinomials of the form ax^ + bx ■{- c. 

Thus, 

Example 1. Factor 8 a;^ - 38 a;+ 35. 

Solution. The first term, 8x% is the product of the first terms of 
the binomial factors. The last term, 35, is the product of the second 
term of the two binomial factors. It is evident that the first term of 
each binomial factor might be ±2 a: and ±.4x, or ±Sx and ±a:; 
also the last terms of the two factors might be ± 7 and ± 5, or db 35 
and ± 1. From these w-e must select those that will produce the 
middle term, —38 x, of the trinomial. Since (+2x) X (— 5) + (+ 4 cc) 
X (— V) = — 38 X, we must take + 2 a: and + 4 x for the first terms, 
and — 7 and — 5 for the corresponding second terms of the two bino- 
mial factors. Therefore, 8 a;^ - 38 a; + 35 = (2 a; - 7) (4 ic — 5). 

Example 2. Factor 6x* ~ bx'^y^ — 6 y\ 

Solution. Take + 3 a;^ and + 2 x^ for the factors of 6 x*, and 
+ 2 y'^ and — 3 2/^ as those of — 6 ?/*. We now arrange them in bino- 
mial factors, so that the algebraic sum of their cross products shall be 
- 5 a:2 1/2. Since (+ 3 a^) X (- 3 7/2) + (+ 2 x^) X (+2y^) = -5x^ y\ 
+ 3 a;2 and + 2 a;^ are the first terms, and + 2 y'^ and —3y^ are the 
corresponding second terms of the factors. .•. 6x^ — 6 x^y^ — 6y* 
= (3 a:2 + 2 ?/) (2 a:2 - 3 y^). Hence, 

II. If the Coefficient of the Highest Power is not Unity 

Arrange the trinomial in descending powers of a common letter. 
Select factors of the extreme terms and arrange them in binomial 
factors, so that the algebraic sum of their cross products shall be the 
second term of the trinomial. 



Exercise 46. 

Factor the following : 

1. 4 x^ + l.S a: + 3; 4 2/2 _ 4 y _ 3; 12 a^ -\- a^a^ - x\ 

2. S + llx-4:X^; Sx^'-22xy-21f; ^o?'x^ + ax-^l. 



FACTORING. 125 

3. 8 m6 - 19 m3 _ 27; 15 a^ _ 58 a + 11 ; 6 a2 ^. ^ ah 
- 3 62; 2 //<2 _ 13 m 71 + 6 n2 ; 3 a^» + 7 a; + 4. 

4 24 + 37a-72a2; 15^:24. 224a: -15; 4-5a;-6a:2. 

5. 6a^»- 19.ry + 10/; % a^ -^^ U xy - Ibf) lb a^ 
-77 a; +10; 24 a,^ + 22 2;- 21 ; lla2 + 34a4- 3 

6. 18-33a;+5ar'; Ga:2_7a,2^_3^. 5 + 32;«- 21a;2. 

7. 24a,'2-29a;y-4y2; ea^^+lQ^ra-yn^y^^m 

8. 2(a;+y)2+5(a;+y)(m + ?i)4-2(m + n)a; 2«2+a;-28. 

9. 2(x+yf-l{x^y){a^h) + Z{a + hf- l^x'j^^x-^. 

10. n{x-yf''-l'6x'^y^{x-ijf''^2x^'^f) 27a2+6a-l. 

11. 8 a2" -f 34 rt» (2: - yY^ + 21 (a: - ;/)2'»». 

55. A trinomial is a perfect square when two of its terms are 
positive, and the third term is twice the product of their .square roots. 
Such trinomials are particular forms of I., and their binomial factors 
are equal. 

Example. Factor 4 z^ + 44 a:y + 121 y\ 

Solution. The first term of each binomial factor will be the 
8C[uare root of 4 x^, or 2 a: and 2x, For the second terms of the bino- 
mial factors tjike the square root of 121 ?/*, or 11 ?/ and 11 y. Since 
the terms of the trinomial are positive the factors are 2 x + 1 1 y and 
2x + lly. Therefore, 

4x« + 44xy4- 121 y2= (2x+lly)(2x+ 11 y) 
= (2x4-lly)^. Hence, 

III. If the Trinomial is a Perfect Square. Arrange the tri- 
nomial .according to the powers of one letter. For one of the equal 
factors, fin<l the square roots of the first and last terms, and connect 
these roots by the sign of the second term. 



126 ELEMENTS OF ALGEBRA. 

Exercise 47. 

Factor the following : 

+ 225&2c2; a6-4a4 + 4a2 

2. 49 m6- 140 m%2 +100714; si x"^ 9/ -126 a^xhj + 49 a^. 

3. 7n}^—2m^ni-n'^; l-10mn + 25m^n^; x'^+2x^i-x^. 

4. {a + hf + 16 (a + ^>) + 64 ; 7?i2 + 18 m + 81. 

5. 4a*a^-20a2^7/ + 25a;42/^ o61aH^c^-16ahcdmn 
+ 4:d^m^'n?; 121 7/2,27^4 - 220 mn'^2^ + 100^2 

6. 225 X^ - 30 a;2 7/2 + ^4 . 4 ^4« _ 4 ^2n ^m _^ ^2m^ 

7. 49 m27i2 + -2^ m n^ + ^71*; ^j2 + ^ + 1. 

8. -^^a^ + ^^^W + \a^h^- a^c + 6a^h^e+9Wc. 

9. 9^:2 _ 3:^7/ + ^7/2; (m - 7O2 + 2 (7?z, - ^i) + 1. 

10. {ci?-af-\.6{a^-a)+ 9; 4 (a; + 7/)2 + Jg- + a? + 7/. 

11. at + 6l — 2 aHf ; 7?i — 2 ??ii + 1 ; ??z2 71 + 77^ 7i2 — 2 77it 77!. 

12. x-\-2 x^y^ + 7/ ; 7?i2 n + a2— 2 a m n^ ; 4 ic+ 1 2 71 a:^ + 9 7i2. 

13. (a + 2>)2"-10(a + &)"c + 25c2; | ^5m_^ _i_6___ 11 J.«. 

56. Example. Factor 8 a;^ - 27 i/^. 

Solution. Evidently (Art. 34) 2 re - 3 y is a divisor of 8 x^ - 27 1/^. 
Dividing 8a:^ l)y2x, we have 4x^, the first term of the quotient. 
Divide 4 rc^ by 2 oj, multiply the result by 2y, and we have Qxy, the 
second term in the quotient. In like manner we find 9 .y^ for the last 
term in the quotient. Hence, the quotient is ^x^ + Qxy + ^ y^. 
Therefore, the factors of the binomial are 2ic — 3?/ and 4x'^+ 6a:?/ + 9?/2. 



FACTORING. 127 

Since the dividend is equal to the prcKluct of the divisor and quotient, 
:i* — 27 y* = (2 X — 3 y) {4 x^ -h 6 xy + d y^). Hence, in general, 

When a Binomial is the Difference of Two Equal Odd 
Powers of Two Numbers. Cunsider the binomiid a dividend, 
and find a divisor and quotient by inspection (Art. 34) The divisor 
and quotient will be the required factors. 

Exercise 48. 

Factor the following : 

1. l-d'i'Sa^; 82^-7297/) 216x^-a^ 

2. 2^,/ - aH^; x^ -1; 243 a^ - b^; a^h^ - m^ 

3. 216 d? — 343 ; 3 a; — 81 a:*. Suggestion. Remove the 
monomial factor 3x first. 

4 a}^ - 1024 6^0 ; 729 x^ - 1728 f\x-^- y-\ 

5. 135a;*-320ar2; 2an-64a&; a;-^-7/-f. 

6. a655-2:6/; 64^6-125^3; a:3n_^« 

57. Example. Factor 729 + a«. 

Solution. Since 729 is the 6th power of 3, 3* + r|2 (Art. 36) is 
a divisor of 729 + a'- Dividing 729 by 3^* we have 3*, the fii-st term 
in the quotient. Divide 3* by 3', multiply the result by a'*, and we 
have 3* a*, the ."^econd term in the quotient. In like manner we 
find a* for the last term in the quotient. Honcc, the quotient is 
3* - 3"^a2 + n*. Therefore, the factors of the binomial are 3* -f a* 
and 3* — 3^ a-* + a*. Since the dividend is efpial to the ]>roduct 
of the divisor and quotient, 729 + (i« = (9 + a^) (81 - 9 a* + a<). 
Hence, in general, 

When a Binomial is the Sum of Two Equal Odd Powers of 
Two Numbers. Let the student supply the method (See Art. 36). 



128. ELEMENTS OF ALGEBRA. 

Exercise 49. 

Factor the following : 

1. d2a^ + 1; 1 + x^; a^ + y^; x^^ + i/^. 

2. a^ + 128; x^ + 729 ^/^ 64 2^6 + y^ 

3. aH^ + 2^10^10; x^ +64/; 1000 a;3 + 1331 ^/^ 

4. a^l8 + yS J 135^5 _!_ 320 :i;2. ^24 _|. ^24, 

5. 2;-5 + 7/-5; j;15 + ^6. ^5^5 +^5^5. ^21 ^ J54, 

6. a54 + 654. 1 4. ^12. ^^n _|_ ^6m. ^-f _^ ^- f ^ 

7. ai2» + &9'" . 32 rj> h^ c^ + 243 a^ ; 1024 a^ + h^^. 

8. 64 a;6 + 729 a^ ; yig a^ + g\ je ; (^2 _ J c)^ + 8 h^c^. 

58. Example 1. Factor 25 a;^ - 64 2/*. 

Solution. The square root of the first term is 5 x^, and of the 
last term 8 y. Hence, since the difference of the squares of two num- 
bers is equal to the product of the sum and difference of the numbers 
(Art. 26), 25 a;2 - 64 3/2 = (5 x + 8 ?/) (5 a; - 8 y). 

Example 2. Factor {6 a - 4)^ - (3a + 4x - 4)2. 

Solution. The square root of each term of the binomial is 5 a — 4 
and 3 a + 4 ic — 4. Adding the results for the first factor, we have 
8 a 4- 4 X — 8, or 4 (2 a + a; — 2). Subtracting the second result from 
the first for the second factor, we have 2a — 4x, ov 2 (a — 2 a:). 
Hence, the factors are 4 (2 a + a: — 2) and 2 (a — 2x). 

Process. 

(5a-4)2-(3a-f-4a;-4)2 = [(5 a-4)+(3 a-|-4 .'c-4)] [(5 a-4)-(3 a+4 x-4)] 
= [5a-4 + 3a + 4a;-4][5a-4-3a-4a; + 4] 
= [8a + 4a;-8][2a-4a:] 
= 4[2a + a:-2][2(a-2a:)] 
= 8(2a + a:-2) (a-2x). 



FACTORING. 129 

A binomial expressing the diiference between two e<iUttl even 
powere of two numbers may often be separated into several factors. 
Thus, 

Example 3. Factor a" — i/". 

Solution. The square root of each term of the binomial is 3* and 
y*. Adding these results for the first factor, we have x^ -}- y^. Sub- 
tracting the second result from the first for the second factor, we have 
x^ — y*. Similarly the factors of x* — y* are ar* + y* and x* — y^. In 
the same way the factors of z* — //* aie x^ + y^ and x^ — y"^. Finally 
the factors of x^ — j/* are x + y and x — y. Hence, the factors of the 
binomial are x* + y*, a:* -I- y*, x* + y\ x + y, and x — y. 

Process, x^' — y^^ = (x^ + y^) (x^ — y^) 

= (x^+y^)(x^ + y*)(x*-y*) 

= (x^ + y^) (a:* + /) (x2 + y2) (x2 -1/2) 

= (a:«+/) {x*i-y*) (xH/) (x + y) (x-y). 

Hence, in general, 

When a Binomial is the Difference of Two Equal Even 

Powers of Two "Numbers. Find tlie square root of eacli term of 
the biuouiial ; add the results for one factor, and subtract the second 
result from the first for the other. 

Notes : 1. Tlie preceding method is a direct consequence of Art. 26. 

2. The above method finds a practical application when it is necessary to 
find the difference l)etween the squares of two numerical numbers. Thus, 

(235)« - '219)2 = (235 + 219) (235 - 219) = 454 X 16 = 7264. 

Exercise 50. 
Factor the following : 

1. a^x^-lP- f ; 10 .r2 - ^y2 . 25 a^x^ - 49 JV- 

2. a,-* - ?/ ; 2r4 - 81 / ; .x-^ - ?/ ; x^^ - f. 

3. a86*-81a^?/«; l-100aH*c2; 16 a^^ - 9 6«. 

4. 9 «2'. - 4 a^- ; i «2 - J /,2 . .^1 __ yl^ 

9 



130 ELEMENTS OF ALGEBRA. 

5. x-^ - /; (^ + hf - (c + df- {X - yf - a\ 

6. a^-[x-yf'Axy^a hf -I) {a ■\- hf - {a -- hf. 

7. (a + l)2-(&+l)2;Xa+ir— C^^-l)2; (753)2 -(253)2. 

8. (24 X + yf -{2Sx- yf; (1811)2 _ (689)2. 

9. {5x- 2)2 -(x- 4)2; (1639)2 - (269)2. 



10. a^"- 1; 729x'^y- xy^; aH - b^ ; a - h. 

11. (2x+ a- Sy -(3-2xf; 64:X-^ - 729 y-^ 

12. (575)2 - (425)2 ; 2 a - 4 2:2 ; 25 a" - 3 6''" ; a:^ - 3/6 

59. Compound expressions can often be expressed as the differ- 
ence of two equal even powers of two numbers, and then factored by 
the foregoing principles. In many such expressions it will be neces- 
sary to rearrange, group, and factor the terms separately. Thus, 

Example L Factor x^ - y"^ -i- a^ - b^ + 2 ax - 2h y. 

Process. 

*2_y«-|-a2-6H2ar-2&i/=:a:2+ 2 ax + a^ - ¥ - 2hy - y^ 

= (x2 + 2 a a: + a2) - (62 + 2 6 y + t/2) 
= (X + a)2 -(b + yy 
= [(X + a) + (^ + b)] [(x + a)-(h + y)] 
= [a + b -i- X + y][a — b + X - y'\ 

Explanation. Rearranging and grouping the terms, in order to 
form the difference of two perfect squares, we have the third expres- 
sion. Factoring the third expression gives the fourth expression. 
The square root of each term of the fourth expression is (x + a) 
and (y + b). Adding these results for the first factor, we have 
a + b + X + y. Subtracting the second result from the first, we 
have a — b + X — y, for the second factor. 



FACTORING. 131 

Example 2 Factor 2xy + I - x^ — y^. 

Process. 2zy + I - x"^ - y* = I - x^ + 2 xy - y^ 

= 1 -(x2- 2x1/4-2/2) 
= 1- (x-yY Art 55. 

= [l + (^-2/)][l-(^-l/)] 

= [l + x-y][l-x + yl 

Example 3 Factor 4a^b^i-4c^d^^8ahcd-(a^-\^b^-c^-(P)'' 

Process. 4 a* />» + 4 c2 </« + 8 a 6 c </ - (a^ + />' - c^ - d^y 
= 4a^b'^ -\-8abcd + 4 c^d^ - (a« + b^ - c^ - d'^y 
= (2a6 + 2crf)2- (a2 -I- 62 - c2 - rf2)a 

= [(2a6 + 2c</) + (a2-f-6*-c2~rf''')][(2aft+2c(/)-(a2 + 62-c2-f7i)] 
= [2a6 -f- 2cd + a* + fc'* - c* - «/2J [2a6 + 2c</ - a^ - &« + c^ + d^] 
= [(a2 + 2a6 + 6-')-(c^-2crf + rf2)][(c2+2crf + f/2)_(rt2_2a6+/^'^)J 
= [(a + ^y - (c - rf)2] [(c + dy - {a - 6)2] 

= [(a+6) + (c-rf)] [(a+6) - (c-^)] [(c+rf) + (a-6)] [(c+d)-(a-6)] 
= [a + 6 + c - (/] [a + 6 - c + dj [a - 6 -}- c + </] [6 + c + rf - a]. 

Explanation. Arranging and factoring the first three terms, we 
have the third expres.sion. The f*quare root of each term of the 
third expression is 2ab + 2cd and a"^ + 1^ — c^ — d^ Adding and 
subtracting these results, respectively, gives the fifth expreasion. 
Rearranging (2a 6 and 2 erf suggest the proper anangenient) and 
grouping the.se terms, gives the sixth expression. Factoring the 
terms of the sixth expression, we have the seventh expression 
Finally, factoring the terms of the seventh expression, we obtain 
the result 

Exercise 51. 
Factor the following expressions : 

1. a2-62-c2-2&c; a^ -{- y'^ -V^ -2 a y • l^-a^-V^^2ah. 

2. 25a:2_2^_(56c_9c2; 2?Jf.2ax-\- a^- y^- 2yz-z\ 

3. 4a;2_i2a:y+ 97/2-81; T^-^x^Uf, 4r*-l+r)r-9z2. 

4. 16^*- r2-f - - J 9rt2_6^4-l-a:2_8a;y-16y2. 



132 ELEMENTS OF ALGEBRA. 

5. x^ — li^ + '^'^^ — n^— 2mx — 2ny\ a* + b"^ — c^ — cf^ 

6. 12 xrj -4 x^ -^J y^ + z^; A x - 1 - 4= x^ + 4 a\ 

7. (^2 _ 2/2 _ ^2)2 _ 4 2/2;22 . (^2« ^ ^2" _ c4m)2 _ 4 ^,2«52« 

8. 4 ^2 _ 12 a a; - 6-2 - fZ2 - 2 c c^ + 9 a2 . 4 a:^ + 9 ?/2 
-16^2 -25 6^2 - 12^?/ - 40 ^^s; Ax^ - W' -2hc-c\ 

9. :i'4 - 25 «« + 8 a2a;2 - 9 + 30 a^ + 16 a* ; 2/2 + 6 6aj 

- 9 62^2 _ 10 ^) ?/ - 1 + 25 ^>2 ; (a4« - 4 a2" - 6)2 - 36. 

10. rz;2« _ 9 ^2 + ^2- _ 2 ^"^"^ _ 6 a & - ^>2. 

11. a;6"-4?/*'" + 12?/2'"^ + 2a3:i.^«_ 9:^2 4. ^6_ 

12. 4 ^2 _ 9 2/2 + 16 ^2 _ 36 ^2 _ 16 ^ ^ + 36 ny. 

13. tt2« -f- ^>2« _ 2 a*^^)'* - c2'» ^ Aj^"* - 2 c"* A;2m^ 

14. 4 ^2 + 9 ^.2 _ 16 (2/2 + 4 ^2) _ 4 (16 2/;2 _ 3 «^.). 

15. a2_,_^5_9^,2 4.i^2. a^-a2-9-2a2^,2 + ^4_^6a. 

60. The method for factoring a trinomial consisting of two trino- 
mial factors depends upon the following axiom : 

5. If the same numher he both added to and subtracted 
from another^ the value of the latter will not be changed. 

Example 1. Factor x^ + a^x^ i- a*. 

Solution. Adding and subtracting a^x"^, we have x* + 2a^x^i-a* 

- a^x^. Factoring the first three terms of this expression, we get, 
{x^ + a2)2 — a^x^. Here we have the difference of two equal even 
powers of two expressions, and it is equal to the product of the 
sum and difference of their square roots. Hence, the factors are 
a* + a a: + a:^ and a^ — ax + x^ 



FACTORING. 133 

Process. 

= X* i- 2a2x2 + a*-a«z« 

= {x^ + a^y-a^x* 

= {xHa^-a x) (xHa'^-a x), or (aHa x+x«) (a«-a z+z«). 

Example 2. Factor 16 a* - 17 a" 6^ + 6*. \ 

Process. I6a*-na^b^-^b* = \6a*-n a^b^+9a^b^-hh*-9a^b^ 
= 16a<-8a«fc2 + M-9a262 
= (4 a^- 62)2- (3 a 6)-^ 
= (4a2 + 3a6-6«)(4a2-3a6-Z>2) 
= (rt + 6) (4 a - 6) (a - ^) (4 a - 6)v 

Explanation Adding and subtracting 9 a* 6* to the expression 
(to form a perfect square), arranging and factoring the terms, we 
have the fourth expression (the difference of two equal even powers). 
Factoring the fourth expression, we get the fifth expression. The 
factors of 4 a' + 3 a 6 - 6* are a + A and 4a - b. The factors of 
4 a* — 3 a ^ — ^^ are a — b and 4 a + b. Hence, 

When a Trinomial is the Prodnct of Two Trinomial Factors. 

Make the trinomial a perfect scjuare by adding the requisite expres- 
sion. Also indicate the subtraction of the same expression. The 
resulting expre8.sion will be the difference of two squares. Take 
the sum of their scjuare roots for one factor, and their diflerence for 
the other. 

Exercise 62. 

Factor the following expressions : 

1. 9a* + 3a2624.4fe*; aHOa^ + Sl; 16 X* -\- 4 aP f ■\- 1/*. 

2. a^ + a^iZ-^-f; Sla^2Sa^a^-\-Ua^; mHm^wa + w*. 

3. 40^+8^*2/^+9?/*; a8 + a*fe2 + 54. 81a* + 36 aHie. 
•1 25a^-9a^}^-\'l6b^; x^ + xy-hy^; a^ + a^f + f. 

5. 16a8 + 8a*&3+92>«; 9a*'{'38aH^+49b^; p^ + pHh 



134 ELEMENTS OF ALGEBRA. 

6. 49 a^ + 110 a?lP' + 81 &^ 9 ^* + 21 ^2 ^^ + 25 /. 

7. m*" + m2» + 1 ; ^*" + 16 :2;2« + 256. 

8. a2-3a&+&2; «4«_6^2nj2m_^j4m. 25m4-44mV+16?i*. 



61. Frequently the terms of an expression can be grouped so as 
to show a common factor. Thus, 

Example 1 . Factor 2am + ^hm — cm — A an— Qhn + 2 en. 

Process. 2am + 36m-cm — 4an — 66n + 2cn 

= (2 a m — 4 a 7i) 4- (3 & m — 6 6 ?i) — (cm — 2cn) 
= 2a (m— 2n) -h Sb(m — 2n) — c (m — 2n) 
= (m - 2 n) (2 a + 3 6 - c). 

Explanation. Grouping the terms of the given expression in 
pairs ; taking the common factor 2 a out of the first, 3 b out of the 
second, and c out of the third, we have the third expression. Divid- 
ing the third expression by m — 2n (the common factor), we have 

2 a 4- 3 & — c. Hence, the factors are m — 2n and 2 a + 3 6 — c. 

Example 2. Factor 12 a^ - 4 a^b - S a x^ + b x^. 
Process. 

12a^- 4 a^b -3 ax^ + bx^ = (12 a^-Sax^) - (4a^b-b x^) 
= 3 a (4a'^ - x^) - h (4 a^ - x^) 
= (4 a2 - x^) (3a-b) 
= l2a + x)(2a- x) (3 a- b). 

Explanation. Grouping the terms in pairs ; taking the factor 

3 a out of the first, and b out of the second, we get the third expres- 
sion. Dividing this by 4 a^ — x^, we have 3a — b. The factors of 
4a^— x^ are 2 a + x and 2a — x. Hence, the factors of the poly- 
nomial are 2a + x, 2a — x, and 3 a — b. 

Example 3. Factor 2mn — 2nx — my + xy -{- 2 n^ — ny. 
Process. 2mn — 2nx — my + xy + 2n^ — ny 

= (2mn — 2nx + 2 n^) — (my — xy + 7iy) , 

= 2n (m — X + n) — y (m — X + n) 

= (m — X + n) (2 n — y). 



FACTORING. 136 

Example 4. Factor -4ax + 4x^ + 4ay-\-4y^-8xy. 

Process. —4ax-^4x^4ay+4y^-6xy = 4[-ax+x^-\-ay+y^-2xy] 

= 4[{x*-2xy-{-y^)-{ax-ay)] 
= 4[{x-y){x-y)-a{x-y)] 
= 4(x-y)[(x^y)-a] 
= 4{x-y)[x-y-a]. 

Example 5. Factor 2am^-2an^-2am-2an-\-2a^2a^4a^n. 

Solution. Remonng the common factor 2 a, we have w* — n* 
-7n — n + a — a*4-2an. Arrange the terms of this expression into 
the groups m* — (n* — 2 a n + a*), and — (m + n — a). The factors 
of the first group are m + n — a and m — n + a. Hence, m^ — n' 
— m — n + a — a2-|-2an = m*— (n* — 2an4-a*) — (m + n — a) 
= (m + « — a) (m — n + a) — (m + n — a). Dividing this expres- 
sion by the common factor, m + n — a, we have m — n + a — 1. 
Hence, the factors of the polynomial are 2 a, m + n — a^ and 
m-n-\-a- 1. Therefore, 2am^-2an^-2am-2an'{-2a^ 
~ 2 a* -h 4 a^ n = 2 a (m + n - a) (m - n + a — I). 

Process. 2 am^ - 2 a n^ - 2 am - 2 an + 2 a^ - 2 a* + 4 a^n 
= 2 a[m^ - n^ - m - n + a - a^ + 2 an] 
= 2a [(my - (n - a)^ -(m + n- a)] 
= 2a[(m + n - a) (m — n i- a) — (m + n - a)] 

=: 2 a (m -f n — a) [m — n + « — 1] . Hence, 

To Factor a Polynomial by Gronping its Terms. Group the 
terms of the polynomial so that each group shall contain the same 
compound factor. Factor each group and divide the result by the 
compound factor. The divisor and quotient will be the required 
factors. If the polynomial has a common simple factor, remove it 
first. 

Note. It is immaterial what terms are taken for the different groups so that 
each group contains a common factor. Tf the groups are suitably chosen the 
result will always be the same, although the order of the factors may be 
changed. Thus, in Example 3, by a different grouping of the terms, we have 

2mn — 2nx-my + xi/ + 2n^ — ny 

= (2mn — my) — (2nz — xy)-{-(2n* — ny) 
= w (2 n - y) - X (2 n - y) + n (2n - y) 
= (2n-y)(m-x + n). 



136 ELEMENTS OF ALGEBRA. 

Exercise 63. 

Factor the following : 

1. a^ ■{- ab -\- ac + be; a^c^ + acd — '2abc — 2bd. 

2. am—bm—an+bn; 4:ax—ay—4:bx+by; af^+a^-{-cr-\-a. 

3. Qax — Sbx — Qay + 3by; pr -\- qr — 2^ s — qs. 

4. ax— 2hx-\-2by-\-4tGy — 4:cx— ay. 

5. 5a2- 5&2_2a + 2&; ^ x^ + Z xy - 2 ax - ay. 

6. 2x^-3^^-4cX-2; a'^x^-a^x^-a'^x^+1; mx-2my 
— nx + 2 7iy\ 4: X — a X -\- 4: a — a^. 

7. x^ + mxy—4:xy — 4:my'^\ 4:a^ + 4:x'^ + 5a — 5x — Sax. 

8. 3a'^—Sac — ab + bc; a'^ x + ab x + a c + ab y + I'^y -{- b c. 

9. ^ ax^ + 3 a xy—5bxy—'Sby'^; mn + np — mp — n^. 

10. rii^ + np — mf — 7i?\ V^y^—2 x^y + ?>a^—21 xy^. 

11. 21 a- 5c + 3 «c- 2&C- 14 &- 35; i^2_5^2/ 
+ 62/^+32: — 6?/; 2:^ — r?;2-|-2;_l, 

62. Example L Factor a;^ + 1/2 + g'-J - 2 a; 1/ + 2 a; 2 - 2 ?/ 2. 

Solution. The expression consists of three squares and three 
double products. Hence, it is the square of a trinomial which has x, 
y, and z for its terms. Since the sign of 2 x 2 is + , and 2xyis—, 
X and z have like signs, while x and y have unlike signs. Hence, 
one of the two equal factors is x — y + z. 

.-. x^ + y^ + z"^ - 2 xy + 2 xz - 2yz= (x - y + z)^. 

Example 2. Factor x^ - 3 x^ y -\- 3 x y^ - y^. 

Solution. It is seen at a glance that the given polynomial fulfils 
the laws stated in Art. 29. Therefore, one of the three equal factors 
is x - y. .'. x^ — 3 x^ y + 3 X y^ — y^ = (x — yy. Hence, 



FACTORING. 137 

When a Polynomial is a Perfect Power of an Expression. 

By observing the exponents, coetticients, and signs of the terms, find 
such expression, as raised to a given power, will produce the polyno- 
mial. This expression will be one of the equal factors. 

Exercise 64. 

Factor the following : 

1. a^ -f 2 a 5 + Z>2 + 2 a 6- + 2 6 c + c2 

2. a2-2a6 + 62_2«c-i-26c + c2. 

3. a^+h^-\-c^-\-2ab-2ac-2bc; 16-f 32a: + 24ic2 

4. a^-15a*x + 90a^x^- 2433:^ - 270a^JT'^+ 405aa^. 

5. a^-2ab + }r^+2ac + c^-2ad-2bc + (l^-2cd-i-2bd. 

6. 27x^i/-l08a?x^y^-64:a^+lUa^xy, 

7. m^—2p x—2 n x-\-n^+]f—2 mn+2m x-\-3?—2 mp+2 np. 

Miscellaneous Exercise 55. 
Factor the following : 

Vott. If the expression has a common simple factor, it should be first 
removed. 

1. 10ar»"-30a:"-40; ar^ + A^+l; \22^if-Z(Sxy-A8. 

2. a;2 _ .56 a; + .03 ; a2 + f I a + 1 ; ^ ^ x - x^. 

3. 3m«n8-3m*7i; 16^8-2; a^-Sl; 6a;54-48aJ*+72a^. 

4. ar»ya-5^^y-^j; aH2-|^a3J-^; 9(a + J)2- 
+ ^xy{a + by-x^f. 



138 ELEMENTS OF ALGEBRA. 

5. a'^—2ax—4:a + x^+4:x; 8— 2x^—4:0^— 2x^; da^+a. 

6. x^ + if ^^ - 3%; ^2« + 16 a" + 63; -i-| a^'" + 
(f ^t" - -¥- ^^") a^'" - 6 a** +^. 

7. m^ — a m — 71 m + a 71 ; a^ + 7 a — 8 ; 4 a^ — 4 2?^ 
-2a + 2b; 49 a^H^"" + 7 (:i;2 + 32/^)a"53«^i _^ 3iz;i2/i 

8. 204-5a-a2; ^8" - -L2_8 ^4n ^ 1|^ 

9. m^ — n^ — mp — np ; a;^ — ic^ + a;* — 2^^ ; a^ — a^ &^. 

10. 380-^-^2. 8 ^10". _ 9 ^5m _|. 1 . (^^ _ ^)4« 

11. 6a.^2_2,__ 77. 12 2:2+108^+168; ^2^2^^^/ 
+ 2/2 — 5^—5?/; I 2^2 — Jg- (5 -m 71 + 3 2/) ^ + ^z'^^^'^ V- 

12. l-Tiy^-Tfy^^ 2 2^2+ 5 ^?/- 3?/2-4aa; + 2a7/. 

13. a2« + (^ _j_ ^) ^«^« + ^/2" ^7/; :r2 + (a + &) ^ - 2 a2 

- a & ; a;- 12 - 7/^ ;. 81 x^ - 22 2^2 ^2 + y^^ 

14. al2& + Z;13. ^3 + ^3_^3^^(^_|_^). a:4'»+c(a + 5)2:2« 

— ah {a — c) (b + c) ] m^ + 71^ + 771 + 71. 

15. a2_2,2_c24.^/2_2(^^_2,c); 4 + 4^'+2«y+^2_^2_2^2^ 

16. x^-V{a + 2h)x-\-ah-^h'^; x^^''+(a-b)x^''-2a'^-2ah. 

17. 250 (m -nf±2; 8 (771 + nf ± (2 m - n^. 

18. 52»c2»2;2'»_ 6"c"a2x" -&~C"2:"+ft2; 49^4_i5^2^2 

+ 121 2* ; (7?i + nf ± (m. - nf. 

19. 4 (771 •— n)^ — (m — n); {m -{■ Tif ± m {m + n). 



FACTORING. 139 

20. x^-b{a-c)x-ac{a-\-b)(b + c); 64:m^-{-12Sm^n^ 

21. 6 2^ + rS3^i/ + 6xf- 6x^f-xi/-12f. 

22. 2^'* -{She + ac + ab)x'' i- 'Sabc(b ■{■ c). 

23. m3 + 4 m /i2 ± 8 n^ ± 2 m^ n ; {711 + 3 n)2 - 9 (m -pf. 

24. x^'' + (a -\- b -c)x^^''- ac - be; {x -^ yf - x 

- y - 6 ; 25 ^ + 24 x^i/ + 16 f, 

25. 9a:*y*— 3 2^^—60,-2 y^; m^— mri — 6 71^^47/1 ip 12 7^. 

26. rrAn^{ab-zf-m^n^{xy^-2zf\ 81a2"_i99a»fe'» 
+ 121 62-; 81 a^" - 99 a2«^,4« 4. 25 fts™ 

27. 18 a:2 _ 24 xy + 8 / ± 36 a; ^ 24 v/ ; 2 m2 j^ 2mn 

- 12 ?i2 - 12 am - 36 a 71 ; a^a^ ± 64 a?iA 

28. 2^^+ 3x'y-282/*+ 28y + 4a:; 2y - ^ay -\- 4.bx 
-\- G a X — 2x —4by. 

29. 7W* 71 — 7?l2?i3 — 7;i3^2 _|_ ^^^ ,^4. ^j^4 _ ^^^^ ^ ^^4 

30. 15 a^J - 16 7/2 - 15 a a: - 8 a; y + 20 a y ; (a - 6) 
(a2 _ c2) - (a - c) (a2 - ^2). 

31. c«^-c2-a2c3rf3 4.rt2; m87l3±512;247n,27l2-3077l7l3 

— 36 71*; aa^ — 3bxy — axy-\-3b/. 

32. 7m2- 7n7i- 69124. 16m -327i; 4 a;^ + 4 ajS - a;2 - ar*. 

33. 4 m8 - 4 7i3 _ 3 71 (712 _ m^) + 2 7?i (71 - 7?i)2 

34 9m»±9a2m7; a.^~16y2 + a;±42^: (x-2xy)^ 

— (a; — 2 a; y) — 6. Query. How many factors in the first part ? 



140 ELEMENTS OF ALGEBRA. 

35. 64.(4: x + yf - 49 (2 a; - 3 7/)2 ; (wt* - iii^ - 5)2 - 25. 

36. m^ + m^ 7i + m 7i^ + iii^ ii^ + m^ n^ + rn? n^ ; (a; — y)^ 

- 1 + 2:y (a; - ?/ + 1) ; (^2 + 4)2-16 a:2. 

37. (m2 + 3 m)2 - 14 (m2 + 3 ??0 + 40 ; (m 7i - t!-)^ 

- m ?i (tw % — 72. — 3) — 9. 

38. :C2H _ ^n _ I ^ ^-n ^ ^-2n. ^-f _ ^-f^ 

39. 14^2 ^3_ 35a3 2;2+ 14a*:r; a;-6 -t/"! 

40. 12:^:5-8^3^2+ 21 ^2^/; 64:cl± 27 ici 
Separate into four factors : 

41. {x-2y)o(^-{y-2 x)t/; (^'"+ 6^"* + 7)2- (^'"+ 3)2. 

42. 4 ^2 (^3 + 18 a &2) _ (32 a^ + 9 ^2 a?) ■ iQ ^^ ,^2 

- (m2 + 4 ?^2 - ^2)2 . (^4m _ 2 rt2-^,2n _ J4«)2 _ 4 ^4m j4» 

■ 43. x^ + ^./ - 8 0^6 ^3 _ g ^9 . ^9m _^ ^sm + 54 _^3»t + 64 

44. m^ — 2 (?t2 + ^2^ ^.^2 _|. (^2 _ ^2^2 

Separate into five factors : 

45. m'' — inP n^+ 2 7?i* n^ —m^n'^; 6 m* 7^2 + m^ n — 6 m^ n^ 

-rn^n^l (a;2m _^ ^2n _ 20)2 _ (^^'^^n _ y2n _^ ]^2)2 

46. ^7'« + ^4/n_;j^g^m__]^g. 16 ^7m_81 ^ »t_ ^g ^4m_^ 3]^ 

Separate into seven factors : 

47, ^12m_^8mj4n_^4mj8n_^ J12n^ 



HIGHEST COMMON FACTOR 141 



CHAPTER XII. 

HIGHEST COMMON FACTOR. 

63. The product of any of the factors of a number is a 
factor of the given number. 

Thu8, since 30 = 2 X 3 X 5, 6, 10, and 15 are factors of 30. 

The product of the common factors of two or more num- 
bers must be a factor of each. 

Thus, since 42 = 2 X 3 X 7, and 66 = 2 X 3 X 11, 2 X 3, or 
6, is a factor of 42 and 66. 

The product of the higliest powers of all the factors which 
are common to two or more numbers must be the greatest 
common factor of the given numbers. 

Thus, since 24 = 2» X 3 and 36 = 2^ X 3^, 2^ X 3, or 12, is the 
greatest common factor of 24 and 36. 

The Highest Common Factor (H. C. F.) of two or more 
algebraic expressions is the expression of highest degree 
which will divide each of them exactly. 

Thus, 3a:«y«i8the H.C.F. of 3a:«j/«, Qx^y\ and 15x<y«2. 

Note 1. Two or more exprcRsions are said to be prime to each other when 
they have no common factor. Thus, 5 a* and 9 b are prime to each other. 

Example 1. Find the H. C. F. of 24 a« 6« c«, 60 a'^'c^y^ 
48a»62c«, and 36a2 6*c«x». 



142 ELEMENTS OF ALGEBRA. 

Process. 24 a^b^c^ = 2^ X 3 X a^ X b^ X c^ ; 

60 a^b^c^f = 2^ X 3 X 5 X a^ x 6^ x c^ X 2^2 ; 
48 a^b^c^ = 2* X 3 X a^ X b^ X c^ ; 

36 aH^c^x^ = 22 X 32 X a2 X 6^ x c^ X x^ 

.'. theH.C.F. = 22 X 3 X rt2 X 62 X c2=: 12a2Z>2c2. 

Explanation. Factoring each expression, it is seen that the only- 
factors common to each are 22, 3, a^, b\ and c^. Hence, all of these 
expressions can be divided by any of these factors, or by their 
product, and by no other expression. 

Example 2. Find the H. C. F. of 2 x^ - 2 a: 3/2, 4 x^ - Axy^, and 
2 x* - 2 a:2 3/2 + 2 x3 3/ - 2 X ?/3 

Process. 2 a:^ — 2 x 3/2 = 2 x (x + y) {x — y) \ 

4 x^ - 4 xy* = 2^ X (x + y) {x - y) (a;2 + y^) ; 
2 x^ -2x'^y^+ 2 x^ y -2xy^ = 2 x (x + y)^ (x - y); 

.-. the H. C. F. = 2 a: (a; + ?/) (a: - 2/) = 2 a; (a:2 _ y^). 

Explanation. Factoring each expression it is seen that the only 
factors common to each are 2, x, x + y, and x — y. Hence, all of 
these expressions can be divided by any of these factors, or by their 
product, and by no other expression. 

Note 2. If the expressions contain different powers of the same factor, the 
H. C. F. must contain the highest power of the factor which is common to all 
of the given expressions. 

Example 3. Find the H. C. F. of 8a^ x^ + IQa^x^ + 8 a^ x\ 
2 a* a;2 - 4 a^ X - 6 a6, 6 (a^ + a xy, and 24 (a2 a; + a x^y. 

Process. 

8a5x2+16a4x3+8a8x4= 23X a^Xx^ (a + xY; 

2a^x^-4a^x-6a^= -2 X a'^X (a + a;) (3a-a;); 
6(a2+aa:)2= 2 X3Xa2x (a + x^; 
24(a2x + aa:2)8= 2^ X 3 X a^ X x^ (a + a:)3. 

The common factors are 2, a^, and a + x. 
.-. the H.C.F. = 2a2 (a + x). Hence, 



HIGHEST COMMON FACTOR. 143 

To Find the H. G. F. of Two or more Expressions that can 
be Factored by Inspection. Separate the expressions into their 
factors. Take the product of the common factors, giving to each fac- 
tor the highest power which is common to all the given expressions. 

Exercise 56. 
FindtheH.C.F. of: 

4. 12 a^lJ^x^ and 18 a^bs^ ; 6 ci^xy, 8 aT^y, dindi ^tii^xy^. 

5. loa^a^y^, ^a^x^f, and 21d^x^7/. 

6. 12 2:8^2 22^ 18 a:*/^^, and 36 a:^^^^. 

7. 20 c8a:V, 8a2x2yl, and \2a^x^yi. 

8. a^hx ■\- al?x and a^h — l/^. 

9. a^y'^ — z^ and ax^y ^h xy -\- axz — hz. 

10. 3a;*+8a:8+4ar*, ?^a^^-ll3^+^3^, and Za^l^7?-\2x^. 

11. Za^x^y-Za^xy-Z^a^y and 3a2a:3-48 a'^ x 
- 3 a2 0^2 4. 48 a\ 

12. x^-{-x,{x^ 1)2 and 2^8 + 1 ; a^" + a:" - 30 and 
a^- _ a:- _ 42; a:8 ^ 27, a:^ _ 9^ and 2 a^ + 5 a; - 3. 

13. a^ — a^y, a^ — xy^, and 7^ — xi^. 

14. a;* -f aj2 y2 _^ y4 ^nd a^ — 2ar^y+ 2xi^ — i^. 

15. 12 (a - hf, 8 (a2 - ^2)2^ and 20 (a* - ?>4). 

16. 8 a: 2; (a; — y) (x — z) and 12yz(i/ ^ x)(y — z). 

17. 4ar»- 12a: + 9, 4a^- 9, and 4a^bx-6a^b. 

18. a?-21f, x^-^xy^^f, and 2 r*- a;y~ 15/. 



144 ELEMENTS OF ALGEBRA. 

19. :r}' — if, (x^ — y^f, and ax^ — 1 axy + iS ay^. 

20. ma^ — mx, 2x^-\-l^ x'^—2^ x, and 4:a?x^—4:a^x. 

21. 24m7i-C?/i+16 ??,-4, 649i2-4, and IG^i^-S^i-f 1. 

22. a;2 + 4 a; + 4, ^3 + g, and 4 ic2 + 2 a; - 12. 

23. 16 ^3 _ 432, a;2 - 6 ^ + 9, and 5 x^ - 13 rr - 6. 

24. rii^—n?, m7i — 7i^+mp—np, and m^—m?n + mn^—n^. 

25. 6 x^ — 9Q X, m a^ y — 8 m y, and 15 ^ a:^ — 60 ^. 

26. a;6«-ll2;3«+30, r^6«_i3^3«4.42^ a^i^j ^6n_^^3n_42. 

27. a;3'' -125, ^2 « _ 10 ^« + 25, and 2 2^2« _ ^ ^n ^ 5^ 

28. Sa^''- 125, 4^2«_25, and 4a:2«- 20^" + 25. 

64. If the expressions cannot readily be factored by inspection, 
we adopt a method analogous to that used in arithmetic for the great- 
est common divisor of two or more numbers. The method depends 
on two principles : 

1. A factor of any expression is a factor of any multiple 
of that expression. 

Thus, 4 is contained in 16, 4 times; it is evident that it is con- 
tained in 5 times 16, or 80, 5 times 4, or 20 times. In general, 

Since a factor is a divisor, if a represent a factor of any expression, 
m, so that a is contained in m, b times, it is evident that it is con- 
tained in r m, r times h, or r h times. 

2. A common factor of any two expressions is a factor 
of their sum and their difference, and also the sum and the 
difference of any multiple of them. 



HIGHEST COMMON FACTOR. 145 

Thus, 4 is contained in 36, 9 times, and in 16, 4 timed. Hence, it 
is contained in 36 + 16, 9 + 4, or 13 times, and in 36 - 16, 9— 4, or 
5 times. Again, 4 is contained in 5 times 36, 5 times 9, or 45 times; 
also, 4 is contained in 10 times 16, 10 times 4, or 40 times. Hence, 
it is contained in 180 + 160, 45 + 40, or 85 times, and in 180-160, 
45 - 40, or 5 times. In general, 

Let a be a factor of m and n, so that a is contained in m, b times, 
and in n, c times. Then (m + 7i) + a = 6 + c ; also, (m — n) -{- a 
= b - c. Again, since a is contained in m, b times, it is evident that 
it is containe<l in r m, r times b, or rb times ; also, since a is con- 
tained in n, c times, it is contained in s n, « times c, or sc times. 
Hence, rm -^ a=irb, and sn -r a = sc. Adding these equations, we 
have {rm -^ s n) ■Ta = rb-\-sc; subtracting the second equation 
from the first, we have (rm — .<* n) + a = rb — sc. The last two 
et^uations may be written (r 7n ± s n) -^ a = r b J^ s c. Therefore, 
rm±sn contains the factor a. , 

Example 1. Find the H.C. F. of 4 a:« - 3 a:^ - 24 a; - 9 and 
8 x» - 2 or^ - 53 X - 39. 

Solution. The H.C. F. cannot be of higher degree than the first 
expression. If the first expression divides 8 x*—2 x^ — b'Sx — 39, it is 
the H.C. F. By trial, we find a remainder, 4 2:* — 5 a; - 21. The 
H.C. F. of the given expressions is also a divisor of 4 x* — 5 a: — 21, 
because 4x* — 5x — 21 is the difference between 8 a:* — 2 z^ — 53a: 
-39 and 2 times 4 a:» - 3 a;* - 24 z - 9 (Principle 2). Therefore, 
the H.C. F. cannot be of higher degree than 4x' — 5z — 21. If 
4 z^ - 5 a: - 21 exactly divides 4 x« - 3 r* - 24 z - 9, it will be the 
H. C. F. By trial, we find a remainder, 2 z^ - 3 z - 9. The 
H.C. F. of 4 z2 - 5 z - 21 and 4 z« - 3 z* - 24 z - 9 is also a divi- 
sor of 2 z* — 3 z — 9, because 2 z* — 3 z — 9 is the difference between 
4 z» - 3 z» - 24 z - 9 and z + 1 times 4 z^ - 5 z - 21 (Principle 2). 
Therefore, the H. C.F. cannot be of higher degree than 2z* — 3z — 9. 
If 2 z* — 3 z — 9 exactly divides 4 z'' — 5 z — 21, it will l)e the 
H. C. F. By trial, we find a remainder, z - 3. The H. C. F. of 
2 z* — 3 z — 9 and 4 z* — 5 z — 21 is also a divisor of z — 3, because 
z — 3 is the difference between 4z* — 5z — 21 and 2 times 2z* — 3z 
— 9 (Principle 2). Therefore, the H.C. F. cannot be of higher de- 
gree than z — 3. If z — 3 exactly divides 2 x* — 3 z — 9. it will be 

10 



146 ELEMENTS OF ALGEBRA. 

the H.C.F. By trial, we find that a: — 3 Ib an exact divisor of 
2x^-3x-9. Therefore a; - 3, or 3 - a; is the H. C. F. 

Process. 4x^-3x^- 24a: - 9 ) 8a;3 ~ 2a;2 - 53a; - 39 ( 2 
2 times the divisor, 8 a:^ — 6 x^ — 48 a; — 1 8 



First remainder. 




4a,2- 5a: -21 


4^2 _ 5 
X times second divisor, 
Second remainder, 


X- 


-2l)4a:8-3a:2-24a:- 9(a:+l 
4a:3-5a;2-2la: 

2x2- 2x- 9 


2 times third divisor. 


2: 


r2-3a:-9)4x2-5a:-2l(2 
4x2-6a:~18 


Third remainder. 


X- 3 


2 a: times fourth divisor, 
Fourth remainder, 

3 times fourth divisor, 




x-3)2x2_3x-9(2x + 3 
2x2 -6x 

3x-9 
3x-9 



Therefore, the H. C. F. = x - 3, or 3 - x. 

Note 1. The signs of all the terms of the remainder may be changed : for if 
an expression A is divisible by — £, it is divisible by + B. Hence, in the 
above example, the H. C. F. is a: — 3, or 3 — a;. 

Example 2. Find the H.C. F. of 4x^ - x^y - xy^ - 6 y^ and 
7x8 + 4x2^/ + 4X^2_3^8 

Process. 

4x8— x2i/—x2/2— 5 2/8)7x8+ 4^2?/+ Axy^— 33/8(7 
4 times first dividend, 28x8+16x2 2/+16x?/2— 12^/8 
7 times first divisor, 28x8— 7x^y— 7xy^—Sby^ 
First remainder, 23x2^+23xp+23/ = 23 ?/(x2+xi/+.y2) 

x2+xy+2/2)4x8— x^y— xy^—5y^(^4x- 6y 
Ax times second divisor, 4x8+4x23/+4x.?/''^ 

Second remainder, —bx'^y — bxy^—by^ 

— 5y times second divisor, —bx^y — bxy'^—b y^ 

Therefore, the H. C. F. = x2 + xi/ + ?/. 

Explanation. Arrange according to descending powers of x, take 
for the divisor the expression whose highest power has the smaller 
coeflficient, and multiply the dividend by 4 (to avoid fractions). 
Since 4 is not a factor of 4 x^ — x'^y — xy^ — 5 



HIGHEST COMMON FACTOR. 147 

given expressions is the H. C. F. of 4 x* — a;'* y - xy^ — b y* and 28 x« 
+ 16 a^»y+ 16 a: y2- 12 i/« (Principle 2). Since 23 y {x^ -{- x y -\- y^) 
is the difference between 4 times the dividend und 7 times the 
divisor, the H. C. F. oi the given expressions is a divisor of it 
(Principle 2). Therefore, the H.C. F. cannot be of higher degree 
than 23y (x^ -^ ^y + y^) If the first remainder exactly divides the 
first divisor, it will be the H.C.F. Since 23 y is not a factor of 
the first divisor, it can be rejected. Therefore, x^ -\- xy + y^ is the 
H.C.F. 

This method is used only to determine the compound 
factor of the H. C. F. If the given expressions have 
simple factors, they must first be separated from them, 
and the H.C.F. of these must be reserved and multiplied 
into the compound factor obtained. Thus, 

Examples. Find the H.C.F. of 54x^y + 60x'^y^ - I8x»y* 
- 132 a:V and 18x«ya - 50 a^^y* + 2x*y*- I2x^y\ 

Solution. Removing the simple factors 6x^y and 2x^y^j and 
reserving their highest common factor, 2x*y, as forming a part of the 
H.C.F., we are to detennine the compound factor of 9a:* - 22x'^y^ 
-3xy*-\-l0y* = A and 9x* - 6x^y -\- x^y^ - 25 y* = B. UA 
exactly divides B, it is the H.C.F. of ^ and B. By trial, we find 
the remainder -y(6x*-23x^y- 3 x !/« + 35 .y«). The H. C. F. of 
A and B is also a divisor of this remainder, because the remainder 
is the difference between B and 1 times A (Principle 2). Reject -y 
from this remainder, since it is not a common factor of A and B, and 
represent the result by D. The H.C.F. of Z) and 2 A (a multiple 
ofi4) is the H.C.F. n\ A ami J5 (Principle 2). This cannot be of 
higher degree than D ; and if D exactly divides 2-4, it is the H.C F. 
By trial, we find a remainder, 153 y^ (3 x^ - x y - 5 y^). The 
H.C.F. of D and 2^ is also a divi.sor of this remainder. Reject 
153 y«, and represent the result by E. The H.C. F. of E and D is 
the H.C.F. of D and 2 A ; and if E exactly divides D, it is the 
H.C. F. By trial, we find that E is an exact divisor of D. There- 
fore, E is the H.C. F. of A and B. Hence, the H.C. F. of the given 
expressions is 2x^y (3x^ - xy - 5 y^). 



148 ELEMENTS OF ALGEBRA. 

Process. 

9x^22x^y^-'3xy^+10y^ = A )9x*-Qx^y+ x^y'^ -25y^=B{l 

1 times the first divisor, 9x* -22xY-3xy^+10y^ 

-6 a;8 y+2S x'Y+'^ x 2/3-35 ?/'» 
= -y{6x^-23x^y-Zxy^+36y^) 

{3x + 2Sy 

6x^-23x'2y-3xy^+25y^ = D) ISx* - 44xY- ^^y^+ ^Oi/* = 2A 

3x times second divisor, l Sx*-^9x^y- 9x'^y'^+l0bxy^ 

Second remainder, 69^:^- 3bxY-^^^xy^+ 20?/^ 

2 times second remainder, 138x3^- 1{)xY-222xy^+ 40y* 
23 y times second divisor, l 38x^y-529xY~ G9xy^+S05y^ 
Third remainder, 459xV-153x2/8-7652/4 

= I53y^(3x^-xy-5y'^) 

3x^-xy-5y^ = E)6xf^-23x^y- 3xy^ + 35y^ = D{2x-7 y 

2x times third divisor, 6x^~ 2x^y — 10xy^ 

Fonrth remainder, -2lx^y+ 7xy^-\-35y^ 

— 72/ times third divisor, —21x'^y+ 7xy^-]-35y^ 
Therefore, K.C.F. = 2x^y {3x^-xy~5y'^). 

Example 4. Find the H. C. F. of 90 x^ y^ - 200 x^ y^ - 10 x^ y* 
and 144 x* 7/ - 64 a: i/* - 16 x^ / - 144 x^ y^. 

Removing the simple factors lOx^y^ and 16 xy, and reserving 
their highest common factor, 2xy, as forming part of the H. C. F., 
arranging according to descending powers of x, we have 

Process. 9x^—xy^—20y^ ) 9a;8— Qx^i/— xy'^— 4y^{^ 

1 times the first divisor, 9^^ — xy^—20y^ 

First remainder, —9x^y +16^^ — —y(9x^—l6y^) 

9x2-162/2)9x3- xy^-20y^{x 
x times second divisor, 9x^—l6xy^ 
Second remainder, 1 5 x ?/2 — 20 i/3 = 5 ^/^ (3 x — 4 1/) 

3x-4?/)9x2-16y''=(3x + 4i/)(3x-4 2/)(3x + 4i/ 
3x + 4y times third divisor, (3x + 4y)(3x — 4y) 

.•. the H. C.F. = 2xy (3x — 4y). Hence, in general, 



HIGHEST COMMON FACTOR. 149 

To Find the H.C.F. of Two Polynomials that cannot readily 
be Factored by Inspection, if the given expressions have simple 
factors, remove them and iirraiige the resulting expressions acconling 
to powers of a common letter. Take that expression which ia of 
lower degree for the first divisor; or, if both are of the same degree, 
that whose first term has the smaller coefficient. If there is a re- 
mainder, divide the first divisor by it, and continue to divide the last 
divisor by the last remainder, until there is no remainder. The last 
divisor will be their highest common factor. The highest common 
factor of the simple factors multiplied by the last divisor will give 
the H.C.F. sought. " 

Notes : 2. If the first term of the dividend or of auy remainder is not ex- 
actly divisible by the first term of the divisor, that dividend or remainder must 
be multiplied by such an expression as will make the fii-st term thus divisible. 

3. Observe that we may multiply or divide either of the polynomials, or any 
of the remainders which occur in the course of the work, by any factor which 
does not divide hoth of the polynomials, as such a factor can evidently form no 
part of the H. C. F. 

Exercise 57. 
Find the highest common factor of : 
1. ar3 4- 2 ar2 - 13 a: + 10 and 2:3 _j. ,2 _ 10 a; + 8. 

3. a^-x^-5x-Ssind 3^-4x^-11 x- 6. 

4 x*-93^+29x^-39x+lS and 4 a^- 27x^+56 a: -33. 

5. 2^ — 5 ax^ 4- 4 a^x and sd^ — ax^ + Za?x^ — 3 a^x, 

6. 2f-l0xf-^%x^y and ^x'^-^xif-^-Za^f-^s^y. 

7. 2a:S-lla;2_9 and 4a;«+ ll2:*4-81. 

8. 18 ar^ + 3 a: - 6 and 18 a:^ _,. 95 ^ ^ 104 a; + 32. 



150 



ELEMENTS OF ALGEBRA. 



9. 15 m^ n^ — 20 m^n? — 65 m^n — 30 m^ and 2amn^ 
-\- 20 am n^ — 16 a mn— 186 a m. 

10. 36 m^ + 9 m3 - 27?/i* - 18 ?7i5 and 27^57^2 - 9 m%2 
-18 m* 72.2 

11. S x^ — 3 X y + xy^ — y^ and 4:X^y — 5 xy^ + y'^. 

12. mnx^ — 82 m 7i a; — 3 //i ?^ and m^ ti^ ^5 _|_ 28 ^^5 ^2^2 

- 9 m^n\ 

13. a;3 - 4 ^2 + 2 ^ + 3 and 2 ^* - 9 ^3 ^ 12 ^2 _ 7 

14. 16 x^-^^xy + 10 7/2 and 6 :?:* - 29 2;3 7/ + 43 i«2 ^2 
-20^?/^. 

15. 2m^n — 10m^n7j'^+ IS m^ny^+ 224:mny^+2d4:ny^ 
and 4 m* 71 — 20 m2 ?i 7/2 — 48 in ny^ + 112 7i 7/*. 

16. 27?2.'*a;4"- 27?i"2;3" - 4 7??,**^2« + 4^";:c" and 6m"a?5n 

- 18 m^'x'^'' + 12 77l"^3« _|. 6 ^«^2n _ (3 ^n^n 

Query. How many factors in this result 1 



65. To Prove the Method for Finding the H. C. F. of any 
Two Algebraic Expressions. Let A and B represent the ex- 
pressions, the degree of A being either 
equal to or higher than that of B. Di- 
vide A by B, and let the quotient be m 
and the remainder D ; divide B by D, 
and let the quotient be n and the re- 
mainder E ; divide D l)y E, and let the 
quotient be r and the remainder zero; 
that is, E is supposed to be exactly con- 
tained in D. 

We will first prove that -B is a common factor of A and B. 

From the nature of subtraction, the minuend is equal to the sub- 
trahend and remainder. Hence, A = mB -\- D, B zz n D + E, and 



Process. 

B)A (m 
mB 
D)B{n 
nD 



E)D(r 
rE 



HIGHEST COMMON FACTOR. 151 

D = r E. Since the division has terminated, E is a divisor of D. 
E is also a divisor of n Z) (Principle 1) and of n D -{■ E^ or B 
(Principle 2). Hence, £ is a divisor oi mB (Principle 1), and of 
mB + D, or A (Principle 2). Therefore, £ is a common factor 
of A and B. 

We must now show that E is the highest common factor. 

Every divisor of A and B is also a divisor of m B (Principle 1), 
and of A —mBy or 2) (Principle 2). Therefore, every divisor of 
A and i5 is a divisor of nD (Principle 1), and of B — nD, or E 
(Principle 2). But no divisor of E can be of higher dej^ree than E 
itself. Therefore, E is the highest common factor of A and B. 

66. Let i4, B, />, E, etc. represent any polynomials. Let m 
represent the H. C. F. of A and B, n the H. C. F. of m and Z), and p 
the H. C. F. of n and E^ etc. Evidently m is the product of all the 
factors common to A and B ; also, n is the product of all the factors 
common to m and Z), and /) is the product of all the factors common 
to n and £, or /> is the product of all the factors common to A, By D, 
and £, etc., which is their H.C.F. Hence, in general, 

To Find the H. C. F. of Several Polynomials. Find the 
H. C. F. of two of them; then of this result and one of the remaining 
polynomials; and so on. The last result found ^vill be the H.C.F. 
of the given polynomials. 

Exercise 58. 
Find theH.C.F. of : 

4- 4 y*, and 2 a;^ + 2 i/. 

2. ai^ + a^-Sa^-^Jx-9, 2-^ x -\- 2^ -{- s^-\- 2x^ + 2x^, 
and 3 + 3a:2_^^4.^^4^ 

3. m" a:3 4. 2 m" ar2 + m" J? + 2 m", 2 x'^ + 6 2^ + 4: s^, 
Za^-h93^+9x-\-6, '6x^- l23^-Sx^-6x, and Sx^ 
-\-2-\-5x-\-S2^. 



152 ELEMENTS OF ALGEBRA. 

4. 2x^-5 x+ 6- 3x\ 3 2;2 + 2 a;3 - 8 :i: - 12, and 

5. Sa^"" - 33 a;2" + 96 a;" - 84, 68 x^" - 92 x^"" - 24a;» 
+ 32 x!^\ a;3" + 11 ^"-6-6 :>j2«, 50 a;" + 20 r?;3« - 60 r2;2H 

- 20, 5 x^" - 10 a^" + 7 a;" - 14, and 3 ^«" - 35 a:^" 
+ 162 2:^" - 372 a;3« + 494 a;2» - 192 x\ 

6. 9 a:2« + 4 X" + 2 2:3" _ 15^ 48 ^n + 30 _ 343 ^" 

- 24 x^"", 8 a;2« + 4 x^'' + 3 a;" + 20, and 2 x^"" + 12 a:^" 

- 94 a;"- 60. 

7. o 3^ — 2 X y^ — 5 x^ I/, 5 xy^— 6 y^ — 3 x^y"^ — x^y 
+ x\ 9 a^ - 8 x^ y - 2i) X7/, S xy^ - 7 x^y - 2 y^ + 3x^ 
10 y^ — x^y^ — 5 x^y + S x!^ — 7 xy^, and x^ — x^y — x^y^ 

- x]^— 2y^. 

Miscellaneous Exercise 59. 

Note. When possible the student should separate the given expressions 
into their factors by inspection. 

rindtheH.C.r. of: 

1. 7^ — xy^ and x^ + x'^y ■\- xy + y\ 

2. x^ — ?/2, {x — ?/)2, of — x'^y, and 2x'^ — 2xy. 

3. 2 x^ — X — 1, X y — y, x'^ y — X y, and 3 x^ — x — 2. 

4. rc6 - 6 X + 5, 2a^+ b-8x + x\ x^ + x^ - 11 x 
+ 9, and 42 2^2 + 30 - 72 x. 

5. a;2_i8a;+45, 22:2-7a:+3, 2^2-9, and 33p-7x-6. 

6. 6a;4"- 3:r3n_^2n_^n_;^ ^^^ 3a^^ - 3a^'' - 2x^'' 

- cr" - 1. 



HIGHEST COMMON FACTOR. 153 

7. a^ - //3^ x^ + ^V 4- 2/^ a:^ - //, x^ + x// + i/, 
a^ + sH^y — x-^ — i/y and r^// 4- ar^y^ + .^3^. 

8. 2ar»+2rt + 4aa;, x^+23^+2x+l, 7b+Ubx 
+ 7 6^-3+ 14 6r2, 3 :r2 - (3 m + n - 3) a: - 3 m - n, 
a^-_ 2 2^2 4- 1, and 2 2:2+ (2;? + ^ + 2) a: 4- 2p + q. 

9. a:* - 27 &8 ic, (2:2 _ 3 5 ^)2^ a a^ - a b x^ - 6 a b^ x, 
and &2:*-4 62a:3 4. 353^, 

10. 4 2:4'»-2 2:3«_^ 3a.»_9 ^nd 2 a:*" + 2:2'^ - 2 2:3» 
+ 3 2:* - 6. 

11. (a + 6) (a - ^/), (« + 5) (6 - a), and (ft + «)^(^^ - ^)^. 

12. 2 63 _ 10 a &2 _^ 8 a2 6, 4 a2 _ 5 « J 4. ^,2^ ^4 _ ^,4^ 
9a*-3a68+3a252_9a8j^ and 3 a^- 3 a26 + aJ2_ j3 

13. 3 2:3'*-3m2.'2'» + 2m2a:''-2m8 and 3a:3"+ 2m2a:* 
+ 8 m3+ 12 ma:3« 

14. (ni — n) {x — y), (m ^n)(j/ — x), and (n — m) {x—y). 

15. 9 2r»+ 3 2:3+ 12a:+ 20 + 2:^ 3 2:2^2 + ^^ _^2 2r» 
+ 12y2+4y + 8, G2:2^_a;6_|.62^^32,4.24anda^»2^ 
+ 3 2:2y + 4 2:2 + 4 y2 + 12 y + 16. 

16. a3-+3a2'»6« + 3a'»62m_^^«^ Sa^^+oJ^'", 4a2"fe2'» 
+ 12 a"fe3m + 8 ft*"", and a2-- 62«. 

17. 2;* — ?n. 2:3 _|_ (^ _ 1) a;2 4. ^ 3. _ ^ ^j^^-l a:* __ ^ ^ 
+ (w — 1) 2:2 ^ ,j 2; _ ^ 

18. 3n2ar»+ 12?;i27i2+ 3n3^- 15mn^x+ \2m^nx 
— Ibmnx^ and 2 m 71 2:3 4. g ^^3 1^2 _ 2 n^ a:3 4. g ^3 ^ 2; 
+ 2 m n^ x^ — ^ m? n^ X — 2 noi^ — ^ m^ nx^. 



154 ELEMENTS OF ALGEBRA. 

19. x'^ — ma^ — mi? x^ — iii^ x — 2 m^, a;^ — 6 ni^ + mx, 
x"^ — 2 m^ — m x, 3 3^ — 7 m x"^ + S m^ x — 2 m^, and x^ 

— 8 7n'^ + 2 7?i X. 

20. 12x^i/-24.3^2/+ 12x^7/, {x^y-xf'f, xy{x^-ff, 
and ^7^y^-2^x^if^ 2^x^i/-^fx. 

21. a^- 2o?h - aV^+ 2h^, a^ + a^h - ah^ - h\ 
«3 _ 3 a 62 + 2 63, a^ -h^, 2>ac-3hc + 2ah- 2V\ 
a^-b\ and 2h^ + a c - he - 2 ab. 

22. a2 _ (^ ^_ c)2^ (^ 4. ^)2 _ j2^ c2 - {a + hf, and a^ 
+ 2a6 + 62+ 2 6c + 6-2 + 2ac. 

23. a^e^ + a^s^-Se, ^4'^+ 5 2:3«_|_ g ^n^ :i:6«_4 2,3"_96^ 
^3«_|. 32:2« + 3a:"+2, a:^'^ - 9 2:2" + 20, ^nd 3a:3n^3^2. 
+ 5 2f^ + 2. 

24. ^ - 2 2^2 + 3 ^ _ 6 an(i ;z;4 - a?3 - ^ - 2 oj. 

25. 4 ^ 2/^ ~ 2 2/3 + 6 ^2 ^ and 4 ^2 ^ ^ ^ x^ — A.xy^. 

26. 35^4-47^2_|_i3^+X and 42^4+41;2^3_9^2_9^_l 

27. m7z,3+2??z7i2+7'/i'/i+2m and 3?i^— 12 ?^3_ 37^2^5^^ 

28. 2m22/5+166m22/2-96m2 2/ + 108m2 and ^mTv^f 

— 144 m 1^ y^—l^m 7? ?/2 — 108 m n^. 

29. 2^4_6^.3^3^2_3^,+l and ^7_3^6+^_4^2+i2;r-4. 

30. 4a;H322;3+36^2^8^ and 8^6_24^4+24a;2-8. 

31. a;^"— 82/3"»aj2«__2;"2/'» + 2/'" and ^2«__4aj«^+42/2m 



LOWEST COMMON MULTIPLE. 155 



CHAPTER XIII. 
LOWEST COMMON MULTIPLE. 

67. A Multiple of a uumber coutains all the factors of 
the j^aven number with higJiest powers. 

Thus, since 24 = 2» X 3, 2« X 3 is a multiple of 24. 

A Common Multiple of two or more numbers contains all 
the factors of the given numbers with highest powers. 

Thus, since 12 = 2^ X 3 and 9 = 3^^, 2^ X 3^ is a common multi- 
ple of 12 and 0. 

The Lowest Common Multiple (L. C. M.) of two or more 
algebraic expressions is the expression of lowest degree 
which can be exactly divided by each of them. 

Thus, 6 a*x»y« is the L. C. M. of 6 a*, x y\ a:», and a«y«. 

Example 1. Find the L. C. M. of 42 a^x y*, 56 a x*y^, 63 a«x*i/«, 
and 21 a* x^y. 

BolutioiL Separating the expressions into their factors, we have 

42 a»a? y* = 2 X 3 X 7 X a*^ X x X y*, 
56 a x*y^ = 2* X 7 X a X x* X y^, 

63 a»x«y« = 3« X 7 X a» X x« X y», 

21 a*x»y = 3 X 7 X a* X x« X y. 

2* X 3* X 7 is the least common multiple of the coefficierts 42, 66, 
63, and 21 ; a* is the lowest power of a that can be evenly divided by 
ich of the factors a*, a, a\ a* ; x* is the lowest power of x that can 
Ihj evenly divided by each of the factors x, x*, x^, x^ ; y* is the lowest 
power of y that can be evenly divided by each of the factors y*, y^, y*, 
y. Hence, the L. C. M. = 2» X 3^ X 7 X a» X x« X y» = 504a«x*y«, 



156 ELEMENTS OF ALGEBRA. 

Example 2. Find the L.O.'M.. oi 6x^-2 x, 9x^-3 x, t5{z^+x y), 
8 (x z/ + y'^y\ and \2 a^ x^ y^. 

Solution. Separating the given expressions into their factors, we 
have 

12 a^-x^y' = 22 X 3 X a2 X a:3 X y^ 
8(^2^ + 3/2)2^23 X 2/2 X (a: + y)2, 

6(a;2 + ar2/) = 2 X 3 X a; x (a; + 2/), 

6a;2-2a;=2 x a; X (3 a;- 1), 

9a;2-3a:= 3 X a; X (3 a;- 1). 

23 X 3 is the least common multiple of the coefficients ; aP- is the 
lowest power of a that can be evenly divided by a^ ; ofi is the lowest 
power of X that can be evenly divided by each of the factors x\ x, x, x. 
Similarly 2/, (a? + 2/), and (3 a: - 1), each affected with its highest ex- 
ponent, must be used as multipliers. 

Therefore, the L. C. M. = 2^ X 3 X a2 x a;^ x 2/^ X (x+yf X (3 a;- 1) 
= 24 a^x^y^ (x + y)" (3 x - 1). 

Example 3. Find the L.C.M. of 4 aa;2 2/2 + n aa;2/2 - 3a t/^ 
a;8 + 6 a;2 + 9 a;, 3 x^ y^ + 7x'^y^-6xy% and 24 aa;2- 22aa:+ 4 a. 

Process. 

4aa;22/2+ llaa;2/2-3a2/2= a X y^(x + 3) X i^x-1), 
a;3 + 6a;2 + 9 a; = xX (x + 3)2, 
3a:32/3 + 7a;22/3-6a;.?/3= x y^(x + 3)X (3a;-2), 

24aa;2- 22aa; + 4a = 2a X (4a;-l) (3a;-2). 

.-. the L.C.M. = 2aa;iy3(a; + 3)2 (4 a;- 1) (3 a;- 2). Hence, in 
general, 

To Find the L. C. M. of Two or more Expressions that can 
be Factored by Inspection. Separate the expressions into their 
factors. Take the product of the factors affecting each with its 
highest exponent. 

Note. The L. C. M. of two or more prime expressions is their product. 
Thus, the L.C.M. of 

(i^ + ab + b^, aS + b% and a^ + b^ is (a^ + ab + b^) (a^ + 62) (^s + J8). 



LOWEST COMMON MULTIPLE. 157 

Exercise 60. 
Find the L CM. of: 

1. 4&3^i/, ^^aT^f, and 63r/^2«. 

2. 24:111 71^3^, Z(Sm^n^2^, and 4871828. 

3. Ua^l^c^, 9aHc2, and ^(Sah^d^. 

4. 12m*n2 2/3, l^mnf, and 24m^?i3. 

5. 12aa;8y*, a:'"^ — ?/-, 2;-— 2a:y + ?/, and rr^ + 2a;^ ^- 2/2. 

6. m^ (a;^ — ^), 71^ (^x — y), and a:"* — y^. 

7. 2a:(a;- y), -ixyix^ --?/), and 62:3/2(2.4. y)^ 

8. 2:2 + a: - 20, 2^^ - 10 a: + 24, and 2:2 _ 2: _ 30. 

9. 2^2+22:, 2:2 + 4a: + 4, 2:243^42, and a^ -\- o x -^^ e,. 

10. 2:* + a2 2:2 + a* and 2-'* — a a:^ — ft^ a; + a*. 

11. a:2 _ 3 2; - 28, 2^2 + 2: - 12, and a^ - 10 a: + 21. 

12. 15 (2:2^ -xy% 21(2^- ar/), and 35 {xy"^ + f). 

13. ar2 - 1, ar^ + 1, and a^ - i. 

14. '6x^-\- llx-^ 0, 32:2 4. 3^, _j. 4^ g^^j 2r^ + 52- + 6. 

15. 2^+{a-^b)x-\-aby a^-\-(a-\-c)x+ac, and a^-{-(b-\-c)x-{-bc. 

16. mx — my — nx + ny, (x — y)^, and Zm^n—?nnn^. 

17. a:2 4. (rt 4 2,) 2, 4 a 5 and ar2 + (a - i) a: - a 6. 

18. ar2 - 1, a^2 4 1, 2:4 4 1 ^nd a:^ + 1. 

19. a:^ + ar*y + a;?/2 -f ?/3^ ^i — x'^y -{- xy^ — if, and ^ 
-\- x^y -xf-f. 



158 ELEMENTS OF ALGEBRA. 

20. 6 aa^+7 a^x^-S a^x, 3 a^ x^ + Ua^x-^ a\ 
and 6x^ + 39ax + 45 a^. 

21. x^ + 5 X + 4:, x^ + 2 X - 8, and a:^ + 7 ic + 12. 

22.* 12 x^ - 23 a:^ + 10 2/^, 4a:2 _ 9^^ _j_ 5^2^ ^nd 30^2 
-— 5xy + 2y^. 

23. a'-^-4&2, a3-2a2Z^+4a&2_8j3^ and a^+2a^b 
+ 4 a 62 + 8 53 

24. a m + <z '/I + & /?i + & rt and ax + ay + hx-^hy. 

25. 8a;2_38a;2/+352/^ 4a:2_:i,z/_53/2 and 2x^-5xy-7y^ 
26. " 2^2 + ?/2, x^ — n:2 2/2 + 2/^ and 2:^ + y^. 

27. 60 a;4 + 5 a;3 _ 5 ^2^ 60 n;2 ?/ + 32 a; 2/ + 4 y, and 
40 a:^ 2/ ~ 2 a:2 2/ — 2 a;?/. 

28. {a + 6)2 - (c + ^)2, (a + c)2 - (& + ^)2, and (a + df 
- {b •+ c)2. 

29. 2^2 -{_ ^ ^ _|_ ^2^ 2^ _ ^ y _l_ y2^ and a:^ + a;2?/2 + 2/*. 

30. 3 a^* + 26 a;3 + 35 a;2^ 6 a:2 + 38 a: - 28, and 27 x^ 
+ 21x^-30x. 

31. 12 a:2n _^ 3 ^n _ ^g, -^^ ar^« + 30 a;2" + 12 x\ and 
32^2n_4Q^„_ 28 

32. « (??/ — n), b (n — m), and — c{m — n). 

33. (a -b)(b- c), (b -a)(b- c), and (b -a)(c- b). 

34. a(& — a:)(a;— ^), b{c—x){x—a), and c(a — a:) (a:— 6). 

35. a:*^ - 2 aj2n _,. 1 ^nd a:*" + 4 x^'' + 6 a^" + 4 a:" + 1. 
Result. a;«" + 2 ar^" - a;^" - 4 rr^" - x^" + 2 a;" + 1. 



LOWEST COMMON MULTIPLE. 159 

68. If the expressious cannot be factored by inspection, find their 
H.C. F., then proceed as before. Thus, 

Example L Find the L. C. M. of 2 z< + a:« - 20 x" - 7 x + 24 
and 2 X* + 3 z» - 13 x2 - 7 X -h 15. 

Solution. The H.C.F. of the expressions (Art. 64) is x^-\-2x-3. 
Dividing each expression (for the other factor) by x* + 2 x — 3, we 
have 2 X* - 3 X — 8 and 2 x* — x - 5. Hence, 

2 x« + x» - 20 x2 - 7 X -h 24 = (x2 -h 2 X - 3) (2 x« - 3 X - 8), 
2 X* + 3 x« - 13 x2 - 7 X -h 15 = (x* -f 2 X - 3) (2 x'^ - x - 5). 

.-. the L.C. M. = (x2 -H 2 X - 3) (2 x2 - 3x - 8) (2 x2 - X - 5). 

Example 2. Find the L.C.M. of x« - Sx^ -f- 19z -h 12, x»- 6x2 
+ 11 X - 6, and x« - 9 x2 -f 26 X - 24. 

Solution. The H.C.F. of the expressions (Art. 66) is x - 3. 
Dividing each of the expressions by x — 3, and factoring the quotients, 
we have 

x»-8x«+19x-12 = (x-3)(x2-5x + 4) = (x -3) (x-1) (x-4), 
x»-6xHnx- 6= (x-3)(x2-3xf2) = (x-3) (x-1) (x-2), 
x«-9x2-|-26x-24= (x-3)(x2-6x-f8) = (x-3) (x-2)(x-4). 
Therefore, the L.C.M. = (x-3)(x-l)(x-2)(x-4) 

= x*-10x»+35x2-50x + 24. Hence, 

To Find the L C M. of Two or more Polynomials that can- 
not readily be Factored by Inspection. Find the H.C.F. of the 
^^iven polynomial!*, and divide each polynomial by it. Then find the 
L. C. M. of their quotients, and multiply it by the H. C. F. 

Exercise 61. 
Find the L. CM. of: 

3. x^-{-2.x-3, :r?-\-Sx^-r:-S, and a:^ + 4 r^ + a: - 6. 



160 ELEMENTS OF ALGEBRA. 

4. ic* — m aj3 — ni^ x^ — m^ x — 2m^ and 3 a;^ — 7 m cc^ 
+ 3 m^ 5:: — 2 m^. 

+ 38^2/3+ 16^2/*- 10 2/^- 

6. a?3 - 9 :c2 + 26 ^ - 24, a;3 - 10 a:2 + 31 2; - 30, aud 
aj3- 112^2+ 38:^-40. 

7. ^-4 -2)3- 4 a;2+ 16 2^-24, 2^3_ 5^2 ^_ 8^-4, and 

a:2 + 2 2: - 8. 

8. 2)3 _^ ^.2 _ 10 a: 4- 8, ic2 + 2 a: - 8, a;2 - 3 a; + 2, and 
ic2- 1. 

9. 6 2;3 + 15 0^2 _ 6 ^ 4. 9 and 9 a;3 4. G 2,2_ 51 ^ 4. 36^ 

10. 2 ar^ - 8 a;4 + 12 a;3 - 8 2:2 + 2 a:, 3 a;5 - 6 2^ + 3 re, 
aud a::3_3^2_|_3^_l 

11. a:* + 5 2^3 + 5 rc2 - 5 2: - 6, a:^ + 6 a:2 + 11 a; + 6, 
and a:^ + 4 2^2 + 2^ — 6. 

12. 2 2)3 + 7 a;2 + 8 X + 3, 2 x^ - 2^2 - 4 2; + 3, 2 a:^ 

+ 3 2:4 + 2 x^ + 3 ^,2 4_ 2 a: + 3, and a?* 4- 2^2 + 1. 

69. To Prove the Method for finding the L.C.M. of any 
Two or more Algebraic Expressions. Let A, B, D, E, etc. 
represent the expressions, F represent their H. C. F., and M represent 
their L. C. M. Also, let a, b, d, e, etc. represent the respective quo- 
tients when Af B, D, etc. are divided by F. Then, 

A =aF, B = bF, D ^ d F, E = e F, etc. (1) 

F is the product of all the factors common to A, B, Z>, etc. The 
quotients a, b, d, e, etc. have no common factor. Hence, their 
L.C. M. is a 6 rf . . . , etc. and the L. C. M. of aF, bF,dF, etc., or 
their equals A, B, D, etc., is ah d ... F. Therefore, M — abde F, 
etc. 



LOWEST COMMON MULTIPLE. 161 

70. Let A, B, D, Ej etc. represent any polynomials. Let A' 
represent the L.C. M. of A and B, P the L.C. M. of N and D, and 
R the L. C. M. of P and Ej etc. Evidently R is the expression of 
lowest degree which can be divided by P and E exactly ; also, P is 
the expression of lowest degree which is exactly divisible by A^ and 
Z>, and N is the expression of lowest degree which is exactly divisible 
by A and B. Therefore, R is the expression of lowest degree which 
is exactly divisible by ^4, ^, 2>, and E, etc Hence, 

To Find the L.G.M. of Several Polynomials. Find the 
L. C. M. of two of theni; then of this result and one of the remaining 
expressions; and so on. 



71. Let A and B represent any two expressions. Let F repre- 
sent their H.C. F., and M represent their L.C. M. Also, let a and b 
be the respective quotients when A and B are divided by F. Then 
A = aF, B = bF, and M = ab F. Multiplying the first equations 
together (Axiom 3, Art. 47), we have AxB = aFXbF=FXabF. 
Therefore, substituting for abF its value M, A B - F M. Hence, 
in general, 

The Product of any Two Expressions is Equal to the 
Product of their EOF. and L.C.M. 



Miscellaneous Exercise 62. 

Find the L. CM. of: 

aZ^a^h-ah^-h^ and a^ - 2 aH - a 6^ + 2h\ 

2. a:4« _ 10 2:2- 4. 9^ ^n 4. 10 r^" 4- 20 ar** - 10 a:"- 21, 
and a;*- + 4 a;8" - 22 ar»" - 4 af 4- 21. 

3. 2:3"-4ar2"3r+ Oa:"^^"'- lOyS" and a:«*-f2a;2nym 

11 



162 ELEMENTS OF ALGEBRA. 

4. s^ -^ Sx'^ + x^ + 3x^ + x + S, 2a^+6x^^2x-6, 
r" + 2x^ + a^+ 2x^ + x+2, and 2a^+3x^+2a^ + 3x'^ 
+ 2;:c+ 3. 

5. X y — b X, X y — a y, i/^ ^ 3 h y -\- 2 h^, x y ~ 2 h^, 
xy — 2hx — ay + 2 ah, and xy — hx — ay -\- ah. 

6. a^" + ^^4" h"^ + a3» 52m _j_ ^2n 53m _^ ^n 54m ^ j5m^ ^nd 

7. .:c2'* - 4 a'-^"*, .:c3'' + 2 a"' *2« + 4 a^'"^;" + 8 a^^ and 

^3« _ 2 ^'« ^2» ^_ 4 ^2m ^« _ 3 ^3m 

8. 27^"" + {2a-3 &)^2n _ (2 52 + 3a&)a;" + 3 ^)3 and 
2a;2"- (3 6-2 c)^?'^- 3 6c. 

9. ^- 2:r2_^ 4^_8^ ^3+2^2_4^_8^ a^-3a^ 
-4:X + 12, and ^^ - 3 a;* - 20 a;^ + 60 aj2 + 64 a; - 192. 

10. x^''-{a-h)x^-ah, x'^''-(h-c)x''-hc, a^^'-x^^'h^ 

- a;6»62 + 58^ and a)2» - (c - a)x'' - ac. 

Find the H. C. F. and L. C. M. of : 

11. 3 o[^ — 7 x^y + 5xy^ — y^, x^ y + 3 xy^ — 3 0^—1^, 

and 3 a^ + 5 x^y + xy^ — y^. 

12. 6^'5 4.i52,4^_4^2^_;l^Q^2^2^y and9^y-27a^22^ 

— 6 a2a;?/ + 18 <x^z/. 

13. 6^3« + ^2n_5^n_2 and 6 2^" + 5 2;2~-3.2J*-2. 

14. a^ -ah + h^ a^ + ah + h\ a^ + a^ h'^ + 6^ a^ + j3^ 
a^ — 6^, and (c^ — h^f. 

15. 2rc2^(6^_io6)a3-30a6 and 3 a;2- (9 a + 15 6) a; 
+ 45 a 6. 



LOWEST COMMON MULTIPLE. 163 

16. a:3n_ 92«J- ^ 26 X-'* - 24 and a:3»_ 122:2-+ 47 ^•"-CO. 

17. (a.-2 + 62)c + (62 + c2)x and {x^ - W) c + {h^ - c^) x. 

18. (2 2^2 _ 3 ,,i2j y 4. (2 m2 - 3 y2) a; and (2 m2 4. 3^2)^ 
+ (2x2+3m2)y. 

19. 2:3«^ 2a'"a:2n_^ a2'"a:" + 2 a^^ and a:^" _ 2 a'"aj2« 
+ a2m^„_2a3'". 

20. a;3 +3a^y + 3a;^^/J + 2/, oi^ - x^y ^ xy^ - 2f, 
sc^ -\- xy -\- y^y and «* + 2^2 ^2 _^ ^ 

21. 20^^x^-h 25a:< + 5a:8~2:~l, and 25 a;*-10 0:24.1, 

22. 2:8* — yS* a^s-y- — i/*", y^(af — f)^, and a:2« 
- 2 a;*y" + 2^'". 

Find the L.C.M. of: 

23. a:*-7a:8_7-2^_^43a.4.42 and ar*-9a:8+9a:3 
H- 41 a; - 42. 

24. ir3+4a:a+6a;+9, «:«+«»- 2 a; +12, and a:2_^_i2. 

25. 4a:«-42:*~29a^»-21 and 4a:«+ 24a;* + 41 2:24.21. 

26. 2a:*-ll2:8+32r24.i0a; and 32:*-142:«-6a;24.5a., 

27. 2:3 _ 6 2-2 4. 11 a; _ e, a;3 _ a:2 _ 14 a; + 24, and 
2:3 + .r2 - 17 2: + 15, 

28. 32:^ + 52:84.525^52.^2 and 32^-2:8+2:2_2._2. 

29. 9a:*+18a:3_.^_92^4.4 and 6a:*+17a:«-10a:+8. 

30. 2 m^ + m2 - m + 3 and 2 m8 + 5 m2 ~ w - 6. 



164 ELEMENTS OF ALGEBRA. 



CHAPTER XIV. 

ALGEBRAIC FRACTIONS. 

72. The expression (a + &) -f (m + n) may be written — 77- . 
It is read the same in each case. 

The second form is called a Fraction ; the dividend is the 
Numerator, the divisor is the Denominator, and the two taken 
together are called the Terms of the fraction, a, b, m, and n may- 
represent any numbers whatever. Hence, 

An Algebraic Fraction is an indicated operation in divi- 
sion. 

A Mixed Expression is one composed of entire and frac- 
tional parts ; as, n 

m-\ 

a 

Note. The dividing line has the force of a symbol of aggregation, and the 
sign before it is the sign of the fraction and belongs to its algebraic value. 



73. Multiplying or dividing the divisor and the dividend by the 
same number does not change the quotient. For, if we multiply the 
dividend by any number, as m, the quotient will be increased m 
times ; if we multiply the divisor by m, the quotient will be dimin- 
ished as many times. A similar method of reasoning may be applied 
to the dividend and divisor. 

A fraction is in its lowest terms when the numerator and 
denominator have no common factor. 

7a^hc 
Example 1. Reduce ^^^ 3,3 to its lowest terms. 

^o 0/ C 



ALGEBRAIC FRACTIONS. 165 

Solution. The H.C.F. of the numerator aud the denominator 

7a^bc X 1 
is la^bc. Factoring, we have 7^2^^ X4ac ' ^^J^^^"^8 ^^^ H.C. F., 

we have Since the terms are prime to each other the fraction 

4ac 

is in its lowest form. 

6 a* + ax- 15 x^ 

Example 2. Reduce ,r: 2 . ^u^^ — TTZz to its lowest terms. 
15 a* + lis ax — 15 x^ 

6a^-\- ax - 15 x' _ (3 a + 5 x) (2 a - 3 a;) 
^^°*^®**' 15a2+ 16aa:-15x2~ (3a4-5a;)(5a-3x) 

_ 2a -Sx 
- 5a-3x' 

Explanation. Dividing the terms of the fraction by their H. C. F., 

we have This result is in its lowest terms, since the 

5a — 3x 

numerator and denominator have no common factor. 

x*-j-x^y-\-xy* — y* 

Example 3. Reduce -7 — ^ 5 -. to its lowest terms. 

XT — x*y — xy* — y* 

x^ + x^y+xy'-y* _ (x* - y*) + (x*y f x y*) 
Process. ^^^y_^^_yA - (^ _ y4) _ (^^ ^ ^^.^ 

^ (x ^ + y«) (x« -y^) + xy (x« + y«) 
l^ + y«) (a:« - ya) - xy (x« + y^^ 

^ (x« + y«)[x«-y« + xy] 
(x2 + y«)[x2-y2_^yj 

x^ + xy-y^ 



When the factors of the numerator and denominator cannot be 
readily found by inspection, their H. C. F. may be found by the 
method of Art. 64, and the fraction then reduced to its lowest terms. 
Thus, 

. T> , 4a»+ 12a«6 -afta- 15!»» . , 

Example 4. Reduce ^ . , ..^ «. -. — i» — Trut to its lowest 

6 a" + 13 a*o — 4 ao* — 15 6* 
terras. 



166 ELEMENTS OF ALGEBRA. 

Solution. 6a^+r3a^b-4ab^- 1568) 4a^+l2aH~ ab^-l5b^(^2 
3 times the numerator, 1 2 a^ +36 a^lSab^ —45 b^ 

2 times the denominator, I2a^-j-26a^b-8ab^-30b^ 

First remainder, lOa^b + bab^—lbb^ 

= 5b(2a^+ab-Sb^). 

2a2 + a&-362)6a8+ 13a26-4a62_ i568(3a + 56 
6as+ 3a2&-9a62 



10 a^b + 5 ab-^- 15 b^ 
10 a^b-{~ 5 ab^- 15 68 



.'. the H. C.F. of the numerator and denominator is 2a2+a6— 36^. 
Dividing each term of the fraction by 2 a^ + a 6 — 3 6^^ we have 
4a8+ 12a26-a62_i5 58 2a + 56 



6a8+ 13a26-4a62- 15 68 3a + 56 



Hence, 



To Eeduce a Fraction to its Lowest Terms. Divide both 
terms by their H. C. F. 

Exercise 63. 

Keduce to lowest terms : 

75ax^y^' ^m^a^y'^^ 4:X^ + 6 xy' 

72 m^n^x^\ mn^{a^-y^f ^ 2^ + Sx+ 1 
24:'m'nix''' w?n{a^ — j^)' x^ — x — 2 



6 m^- 11m -10 20 (a^ - if) af"!/^" 

6 w2- 19 m +10' 5 2:2 +5 ^2/ + 5^/2' ^imyn+i- 

3 m^ + 23 yyi - 36 3 772^ + ^m^n + ^Trv^Ti^ 

4 m2 + 33 m - 27 ' m* + m^Ti - 2 m27i2 * 

^ in + Zmx x^ — {a -\- b)x + ab 
4 ml — 4 mt 2:2 ' a:2 + (c — ^t) 2: — « c ' 



(m + w)2 — a:2 ^.s _ 3 ^2y + 3 2; y^ _ -^yS 
m a: + 71 a: — a:2' a:^ — x^y — xy^ -\- y'^ 



ALGEBRAIC FRACTIONS. 167 

cr^ — (y + mf ac — ad — he -\- hd 
x^ -\- X y ■{- m X* a^ — b^ 

0.^4- (g + b)x+ ah 27a + a^ 

a^Jt\a-\-c)x+ ac' 18a-6a2+2a8' 



10. 



11. 



12. 



(a _ 6)2 _ c2' (6 4- a;)-^ - {a + c)2 

m^ — m* n — m n* + 71^ a a^ — 6 ic*"*"^ 
m* — m^ 71 — m^ ?i2 -|- ??i ti^' a^hx — h^a^ 



a8+3aH+ 3a62+268' 48 ar* + 16 a:2 _ 15 ' 



7*. Example. Reduce :7^ :r— ^ — to a mixed 

expression. 

ProcesB. 

2x - 3y ) 4x« - 16x2y + 29a;i/2 - 22y« ( 2z2 - sxy -f 7ya + 5-^^^^4— 

-10x2y + 29x1/2 
-10xgy-H5xy« 

14xy^-22y« 

14xy«-21y« 

- .y» • 

Explanation. Dividing the numerator by the denominator, we 
have 2 «* — 6 xy + 7y'' for a quotient, and a remainder of — y«. As 
— y* is not exactly divisible by 2x - 3y, we indicate the division and 
atld the result to 2 x* — 5 x y + 7 y*. Hence, 

To Reduce a Fraction to an Entire or Mixed Expression. 
Divide the numerator by the denominator. 



168 ELEMENTS OF ALGEBRA. 

Exercise 64. 

Eeduce to an entire or mixed expression 

1 + 22J 2x^-x^- 9x^+14: 
' r^^^J^' 2x^- x-S 

x^-2x^ 6 a3 - 13 o2 + 6 a - 6 
z. - 



3. 



5. 



-x+ 1' 3a^-2a + 1 

x^ + ax^-3 a^x- S a^ ^^3 4. 2 a:^ - 12 a: - 13 
x-2a ' a;2 + ^ - 12 

x^ — 2 x^y^ + f/ x^-\-(m + n+l)x + mn+a 
x^ + 2 xy + y^ ' x + n 

6a^-5x^+ Ix-b a^S" - a:^"?/" + x^'y'^'' - y^ ^ 
2x+l ' ^'" - 2/" 



75. Every expression may be considered as a fraction whose 
denominator is unity. Thus, a = - ; a^b — c^ = . 

2;2 _ y2 _ 5 

Example. Reduce x -\- y to fractional form. 

^ x-y 

a:2 - «2 _ 5 X + y x^ - y^ — t) 
Process. x-\-y — ~ ^ ^ 



{X- 


^V) 


X - 
X (rc- 


-y 


x' 


ar2- 


1 


X(x- 




-5) 



5 
X —y ~ X — y' 

Explanation. Writing the entire part in the form of a fraction 
whose denominator is 1, and multiplying both terms of it by x — y, 
we have the third expression. Since the sum or difference of the 
quotients of two or more expressions divided by a common divisor, 
is the same as the quotient of the sum or difference of the expressions 
divided by the same divisor, we have the fourth expression. Uniting 
like terms, we have the result. Hence, 



ALGEBRAIC FRACTIONS. 169 

To Reduce a Mixed Expression to the Form of a Fraction. 

Multiply the entire puit liy the denouiinator ; to the product annex 
the numerator; unite like terms and under the result write the 
denominator. 

Notes : 1. In the above example, since the sign before the dividing line 
indicates subtraction, we most subtract the numerator, x^ — y^ — 5, from 
{X + y) (x - y). 

2. If the sign of the fraction is — , and the numerator is a polynomial, it 
will be found convenient to enclose it in a symbol of aggregation before annex- 
ing it to the product. 

Exercise 65. 
Reduce to fractional forms : 

1. a — X -\ ; — ; - -\- I] a + r- . 

a -\- X m — n a — b 

2. „,._,„, + ,.-_^^; ^^j;±_^^ -(.-,). 

, « — m^n^ a . „ . . i m* — 1 

3. m n H ; m^ + m^ + ?>i -f 1 



m n m — 1 

4. ic + 1 4- ; ; -3-^ — 3 + 1 ; w (a: + ?/) + — — 

7/r — ar x -r t 

2 n (3 w2 + ?i2) wi* + w* 
6. m + w ^ — , — ^2 — ; (w + ny — r^ • 



7 a^m _a^,^«^ y8»_ 



a^m ^ sTir ■\- 7/2" 



8. .^^-,2^/+y»--^'"-'^-^"-'^°r-r-^'--^" 



170 ELEMENTS OF ALGEBRA. 

76. It may be sliown by multiplication (Art. 22) that : 

{+a)i+b) =ab; (-«)(-&) = ab. 

(+a)(+b){-\-c) =abc\ {-a){-b){+c) =abc. 
(+a){+b){+c){+d) = abed; {-a){-b)(-c){-d) = abed, etc. Hence, 

In an indicated product of any number of factors, all the signs of 
any even number of factors may be changed without changing the value 
of the product. Thus, 

(x-y) (:y-z) = (y-x)(z-y); 
(w -x) (x- y) (y~z) = {X - w) (x - y) (z - y), changing the 
signs of the first and third factors. 

Note. In order to multiply a product containing several factors by a given 
expression the student must be careful to multiply only one factor of that 
product by the expression. Thus, in order to multiply both terms of the 

fraction ; — '-— — ~- by a, we must multiply either a-\-b or c + d and 

m -\- n or X -}- y hy a. 

77. It is often convenient to change the order and the signs of 
the terms of the numerator or denominator, or both. Thus, 

Change the order and the signs of the terms of the numerator and 
denominator of the following fractions : 

b — a m — n 

1. -• 2. 



y -X ' (jc — h) {x — m) 

Solutions : 1. Multiplying both terms of the fraction by —1, we 

have 

b — a __ (b — a) X —I _a — b 

y—x~{y~x) X —l~x—y' 

2. Multiplying the factor x — m and the terms of the numerator 
by — 1, we have 

m — n _ (m — n)x— 1 n — m 



(c ~b){x- m) (c - b) [(x - m) X - 1] (c--b)(m~x)' 

Multiplying the factor c — b and the numerator of this fraction by 
— 1, and since adding a negative quotient is the same as subtracting 
a positive quotient, we have 



ALGEBRAIC FRACTIONS. 171 

n — m _ (n-m) X — 1 + (n - m) 

(c-b)(m-x)~'^i(c-b)X-l](m-x)~~ (b-c)(m-x)' 

Change to equivalent fractions having the letters arranged alpha- 
betically, and the first letter of each factor in the numerator and the 
denominator, positive : 

x — m (b — a){c — a) 



3. 



{b-a)(a~c)(y-x) " (d - a)(c - b)(n - m) 



Solutions : 3. Multiplying the numerator and the factor y — x 
by — 1, we have 

X ~ m _ m — X 

(b -a) (a- c) (y-x)~ (b - a) (ti -c)(x-y)' 

Multiplying the numerator and the factor b — a of this result by 
— 1, we have 



(6 -a) (a- c) (x - y) (a- b) (a ^ c) (x - y) 

4. Multiplying the factors c — a and n — m, b — a and c — 6 by 
1, respectively, we have 

(6 -a)(c- a) (a - b) (a - c) 



{d - a) (c -6) (n - m) (d - a) (b -c)(m- n) 

Q;„,n„rlv (a-h)(a-c) (a - b ) ( a - c) 

oimuariy, ^^ _ ^^^ ^^ _ ^y ^^ - n) ' (a - d) (b -c)(m-n) 



(d — a) (c — o) (n — m) (a ^ d) (b — c) (m — n) 



Exercise 66. 

Change each of the following fractions to four equivalent ones 
with respect to the signs of letters : 

^ 7W — 71 _ a — Jj m VI -\- n ^ a 

a — b^ m + n — x' a — h -\- x' m — n -\- a* 



172 ELEMENTS OF ALGEBRA. 

Change the following fractions to equivalent ones having m and n 
positive in both terms : 

m — a a + m — X a + b — n 
b — n ' b — m — y^ a — b + m 

X — m X — m {a — m) (b — m) 

y — n^ (y — m)(z — n)' (c — m) (x — n)(y — m) ' 

Change the following fractions to equivalent ones having the let- 
ters of the terms arranged alphabetically and the first letter of each 
factor in the denominator positive : 

. 2x-?> — y 3 — c + ft 



[m. — a) {2 X — b) (b + a)' (y — x) (m — n) (a — c)' 



^ (x — m.)ba 

5. r^ T7-7T- 



xy mn{c — b) (b — a) {c — a) 

{jj-x)yx 



cb a {b — a) {z — y) (c — a) {y — x) (n — m)' 



78. Fractions having a common denominator are similar. 

Thus, ^, — =-, and — ^ are similar. 
ab ao ab 

2x 3 5 n^ 

Example 1. Reduce r — 5, «, and -. — i to similar fractions 

having the lowest common denominator. 

Solution. Evidently the lowest common denominator is 20 in^n^x^f 
the L. C. M. of 5 m^, mn% and 4x^. Dividing 20 m^n^x^ by the 
denominator of each fraction, and multiplying both terms of each 
fraction by the quotient each by each, we have 



ALGEBRAIC FRACTIONS. 173 

2x 2x X 4n^x^ 'Sn^x* . 

6 m« ~ 5 m* X 4 w» x« ~ 20 m^n*xfi' 

_3 3 X 20ma:*_ _6()mx*_. 

5na _ 5 n» X 5 m^ n» _ 25 m ^ n^ 
4x^~4x^ X 5 m^n* ~ 20 m^n^icfi* 

Example 2. Reduce ^aJ^s^^^is * a:«!4x-5 ^ ^'^^ x44^r + 3 

to similar fractions with lowest common denominator. 

Solution. The lowest common denominator is (x — 3) (x — 5) 
(x + 1) (x + 3), the L. C. M. of the denominators. Dividing the 
L. C. M. by the denominator of each fraction, and multiplying both 
terms of each fraction by the quotient each by each, we have 

x-1 (a;-l)X(a:+l)(x+3) (x + 3) (x^ - 1) 

x^8x+15~ (x-3)(x-5) X(x+l)(x+3) ~ (x + l) (x-5)(x2-9)' 

x+3 (x-f-3) X (x-3)(x+3) (x + 3)g(x - 3) 

xa-4x-5~ (x-5)(x+l)X(x-3)(x+3)~(x+l)(x-5)(x2-9)' 

x-5 _ (x-5)X(x-5)(x-3) (x - 5)^ (x - 3) 

x«+4x+3 ~ (x+3) (x+ 1) X (x-5) (x -3) ~ (x+ 1) (x-5) (x^9)* 

Hence, in general, 

To Reduce Fractions to Equivalent Fractions having the 
Lowest Common Denominator (L. CD.). Find the L.C.M. of 

the denominators. Then multiply both terms of each fraction by 
the quotient of the L. C. M. divided by the denominator of that 
fraction. 

Kotef : 1. When the denominators have no common factors, the multiplier 
for both terms of each fraction will be the product of the denominators of all 
the other fractions. 

2. In all operations with fractions it is better to separate the denominators 
into their factors at once; and sometimes it is also convenient to factor the 
numerators. 

3. It will he observed that the terms of each fraction are multiplied by an 
expression which is obtained by dividing the L. C. D. by its own denominator. 
It is not necessary to state how the multiplier is obtained in every expression. 



174 ELEMENTS OF ALGEBRA. 

Exercise 67. 

Eeduce to similar fractions with L. C. D. : 

a m X ahe 1 2 5 
b' n' y^ mn^ ah' ac he' 

m + n m — n n a a — n n 

ah be a c b in da 

m + 2 n 2 m. — 3 n 5 m — n 
Sm ' 6 n ' 10 m 7^ * 

1 x + 2 x-2 



6. 



x'^-V x^-V x-2' x^-x-2 
m •— n m + 2n m^ 1 



m + n' m — n m^ — 7i^' a + b' a — h' a^ + b'^ 
8m + 2 2m-l 3m +2 



ft 

m — 2 ' 3 m — 6 ' 5 ??i — 10 



_ xy m — n 



8. 



Tfix — rmy •\- nx — ny' 2 oc^ — 2xy 

m n a 

m + x' m^ + x^' m^ — mx + a^ 

X y m 



x^ — xy + y^' x^ + X y + y^' ic* + x^y^ + y^ 
^^ x — y x^ + ?/2 y x^ + y^ 



11. 



x^ + xy + y^' x^ — y^' x — y^ 5 



12. 

13. 



ALGEBRAIC FRACTIONS. 175 

b X 



(a -f xf - 62' (5 _^ 2^)2 _ ^2» a:« - (a + 6)« 
ate 



(c-a)(6-c)' (a-c)(c-6)' {c-a)(c-l) 

a _ g X -1 -g 

Sestioii. (^_^)(^_c) - f(c-a) X -l](6-c) - (g-c)(6-c)* ** 

3a 4a 5 am n y 

3"=^' o^' (a -3)2' n^' wrn:' r=^2 

12 3 4 



14. 
15. 



(2-a;)(3-aj)' (a:-l)(2-a:)' (a:-2)(l-^)' (a:-l)(a>-2) 



1 1 

Suggestion, (g - x) (3 - a:) = (x - 2) (x - 3) = ^*^- 



16. 



17. 

18. 
19. 



(m — x) (a; — n) * (a: — ?w ) (a — a;) ' (a: — a) (n — a;) 

1 + a: 2 -i- a; 

(l-a;)(2-a:)(a;-5)' (x - 1) (2 - aj) (3 - a:) (5 - a;) 

a;- 3 a;- 2 g2 + 4 2 

4^=^' a^+a;-6' 9-6a;4-ar»' ir2-a:-6' 

^ ar^"* + 1 A2m_ 1 

a:*"- 1' a^'" + 42:2.-^_ 3' a^« + 2 ar»'" - 3 ' 



79. Example 1. Find the sum of t , -\^ and -. 

o a n 

Solution. Multiplying the terms of the first fraction by dn, 

of the second by bn, of the third by 6rf, and adding the results 

(Arts. 32, 14), we have 

a c m _adn hen hdm _adn-\-hcn + bdm 
h'^ d'^ ~ii~ h(U'^ bd^'^ bdTk "^ hdik ^~ " 



176 ELEMENTS OF ALGEBRA. 



m a 

Example 2. Subtract - from t 

n 



Solution. Multiplying the terms of the first fraction by 6, of the 
second by n, and subtracting (Art. 19), we have 

a m an hni an — hm 



h n bn bn b n 

2a-3h ^ Sx-2b 
Example 3. Subtract — ^^ irom — ^ • 

Solution. Keducing to similar fractions with L. C. D., we have 

3 a:- 26 2 a - 3 6 _ 6ax - 4ab 6ax-9bx 
3x 2a ~~ 6ax 6ax 

6 ax — 4ab — (6 a x — 9 b x) 
~" 6ax 

■'J _ b(4:a-9x) 

~ Hax 

T^. -11 p 2x — my Zx — ny 

Example 4. Find the sum ot a -I '- and b 

m n 

Solution. Uniting the entire parts, and reducing to similar 
fractions, we have 

/ 2x — my\ (^ 3 a!; — nwN , (2x—my)n C3x—ny)m 

[a+ -)+ [h '-) = « + &+ ^^-^ ^^ 

V m / \ ^ J '^^ ^^ 

(2x — my)n — (3x — ny)m, 

= a-i-b-\ '- ■ '- 

mn 

, (2 n - 3 m) a; 

mn 

Note 1. If the sign of a fraction is — , care must be taken to change the 
sign of each term in the numerator before combining it with the others. In 
such case the beginner should enclose the numerator in parentheses, as shown 
in the above work. 

2^; g x+2 a;-|-l 

Exampi^e 5. Simplify ^2 + 3^ + 2 " x^--2x-3 " x^-x-6 ' 



Process. 



ALGEBRAIC FRACTIONS. 177 

lar-6 x-j- 2 x -\- I 



x2 + 3x+2 x^-2x-3 x^-x-6 

2 (x - 3) x + 2 x+ 1 

= (x + l)(x + 2) ~ (x + 1) (x - 3) ~ (x 4- 2) (x - 3) 

_ 2(x-3)X(a?-3) (x-t-2)X(x4-2) _ (x4-l)X(a?+l 

"(x+l)(x+2)X(x-3) (x+l)(x-3)X(x+2) (x + 2)(x-3)X(x+l) 

_ 2 (x - 3)^ - (x -f 2y -(x+ 1)3 _ 13-18X 

(x + 1) (x -h 2) (x - 3) ~ ix+ 1) (x + 2) (x - 3) * 

Notei : 2. In finding the value of an expression like — (x -f 2)*, the be- 
ginner should first express the product in a parentheses and then remove the 
parentheses as above. 

3. Sometimes it is better not to reduce all the fractions to the L. C. D. at 
once. Thus, 

14 6 4 1 

Example 6. -• — 5 -I rTT + 



x — 2y x — y x x + y x + 2y 

1 1 4 4 6 

+ iTT-zr- - z — - - zr-r-7. + z 



x — 2y x-h2y x — y x + y x 

x + 2y x-2y 4 (x + y) 4 (x - y) 6 

(x-2y)(x+2y)'^(x-|-2y)(x-2y) (x-y){x-\-y) (x+y)(x-y)"^x 

2x 8x 6 



~ x^ - 4 y2 x2 - y3 

_ 2 X (x^ - y^) _ 8 X (x« - 4 y«) 6 

- (x«- 4 f) (x^-y^) " (xa - y2) (x3 - 4 y^ + X 

_ 30xy''-6x« 6 

-(x»-4y«)(x«-y2)'^x 

_ (30xt/'-6x»)x 6 (x2 - 4 y2) (x^ _ yS) 

- (x2 - 4 y3) (x2 -y*)x'^ X (x^ - 4 y^) (x^ - y«) 

24 y* TT . 

= x(x«-4y^(x»-t/) - Hence, m general, 

To Add or Subtract Fractions. Reduce to similar fractions 
with L.C. D.; add or subtract the numerators, and divide the result 
by their L. C. D. 

12 



178 ELEMENTS OF ALGEBRA. 

Exercise 68. 

Simplify : 

2a — 5 S a — 11 b + c a + c a — b 
12 a "^ l8 ' T^ "^ T6 97" ' 

X 2x ^x^^ xy xy^ xP"y^ 

^ m n fm + n a + b\ /m — n a — b\ 
' ab ac b c'' \ n a J \3?i ^a )' 

3 + ^^ 4-am a /5 4 3\ /I 2 3\ 

4. + + ^; +-)+ )• 

n an 6n \m n xj \m n xj 

5a-b 7a + Sb _ /2a a - b \ 
2b "^ 6^ \b "^ 3 & y * 

6. (^,. + ^|) + (3m-^)-(4m + ^). 

^ a^—bc ac—b^ ab — c^ 2 a^—b^ b^—c^ c^—a^ 
i c ac ab ' a^ b^ (? ' 

8. (m + n ^ ) — ( 2m — 371 + — V 

\ mxj \ nxj 

\5 a; Zy 'omj \6 x 10 3/ 7 m/ 
a + & b — c c — a ab'^ — b(? — c 0? 



11. 



5 ab c 

1 1 :r4-2 x-2 3 



oj— 5 ^—4' 2;— 2 a;+2' 2m(m— 1) 4m(?w— 2)" 





ALGEBRAIC FRACTIONS. 


179 


12. 


2a;/i — 3&71 2a7n + 'Sbn 1 


1 


'6vin{ia—7i) 3 7/1 7i (?/i + 7i) * ic^*--4ic + 4 


"x-'^+x-G* 


13. 


x + y J^ — y ' x—m X'\-m x 


14. 


1 m+3 -7 2 3 

... 1 •> "T A 2 10> .J.^ " 


2m-3 

1 A ™2 1 • 



m n 2 mn 1 (a + 2xy^ 

m -i- 71 m — 71 ir? — r?' a — 2x a^ — S a^' 

2: 1 1 11 



16. 



xy— if' x — y y' m^— (n + xj^ a^—(m + ?i)^ 



x^+'Sx^f+y* a^xy-\-ii^ 2 3? x + y 

x^ — ^ X — y ' x*—y* x^+x?y+xif+y^' 

,Q x + 4: , x-hS x+2 



3?+ ox-{- 6 a? + 6 x+ 8 3^+7 x+ 12 

1 mn m — 71 

7n + 71 TTi^ -\- n^ 7n^ — m 71 -\- 71^ ' 

90 1 1 X X 

7n + X m — X " (m + xf (tti — x)^' 

21 __i L_ + ^ 4. _ ^ 



8-8ic 8 + 82:^4 + 40^2^ 2 + 2a;* 

22 24a; 3 + 2a : 3 - 2 a ; 

9-12a; + 4ar» 3 - 2 a: "^ 3 + 2a; ' 

00 « + & ?^ + r- r + ^ 

(6 - c) (c ~ a) ■*" (c - a) (a - ft) ^ (a - I) (h - c) 



180 ELEMENTS OF ALGEBRA. 



24 ^+^y , x-^2y x^y 



4.{x+y)(:y+2y) {x+y){x-VZy) 4.{x+2 y) {x+Z y)' 
^^ he ac , ah 



(c — a) (a — b) (a — b) (b — c) (b — c){c — a)' 

26 5(2^-3) 7x _ 12(3a^ + l) 

• 11(6^-2+ ^-1) 6 2;'^+7ic-3 ll(4:X^ + Sx+3)' 

x _ y x^y+xy^ x^ + 'f ^ — y^ 

x^+y^ Q^—i/ x^ — y^ ' a^—xy-\-y'^'~x^+xy-\-y'^' 

28. ^" + ^* 1-^ 



a;3 + ^2 _ 49 ^ _ 49 2:2 _ e ^ _ 7 
29 ^ + ^' a; 4- & ^ + 0^ 



30. 



{a — b) (a — c) (b — a) (b ~ c) 



Suggestion. In finding the L. C. D. it is better to arrange the 
letters alphabetically. Thus, 

& a b a X —1 

+ VI — wi — X = / — jaT N + m — \t:^ — TT7I — \ = etc. 



{a-b){a-c) ^ {b-a){b-c) ~ (a-h){a-c) ^ [(b-a) X -l](b-c) 

x^+2x+4: oc^—2x-\-4: x-2a 2{o?-Aax) 3^ 

x-\-2 2—x ' x-\-a a^—x^ x—a 

32 1 1 1 . 1 , ^ 

(m-2)(x'+2)^(2-m)(^ + m)' 2a;+l 2aj-l 

4a: 2 a: -3 a?2 

+ -. T—o\ — ^--o ...... + 



1-4^2' ^_l_4 :x:2_4^+i6'^^64 
33. 7 iw T + 



(a — &) (a — c) {b — a)ib — c) (c — a) {c — h) 



ALGEBRAIC iRACTIONS. 181 

Q C 

80. Example 1. Find the product of r and ^ . 



a , c 



Solution. Let i = a:, and 3 = y. Multipljring both members 

of the first equation by h and both members of the second by d (Art. 
47, Axiom 3), we have a = bxy and c = d y. Multiplying these two 
equations together, we have ac = bdxy. Dividing both members 
of this equation by 6 d (Art. 47, Axiom 4), gives 

ac _ a c 

^ = xy. Buta:i/ = ^X^. 

a c ac 

Therefore, T ^ ^ = r-, . Hence, in general, « 

To Multiply a Fraction by a Fraction. Multiply the numera- 
tors together for the numerator of the product, and the denominators 
together for the denominator of the product. 

Notes : 1. Similarly, we may demonstrate the method when more than two 
fractions are multiplied together; also, for fractions whose terms are negative, 
integral, or fractional. 

2. Since an entire or muted expression may be expressed in fractional form, 
the method above is applicable to all casesl Thus, 

^a m ^a am a / ,n\ a^/>».n\ am , an 

r. o Ti'j.u J . -4x3-16x4-15 x2-6x-7 

Example 2. Fmd the product of ^ q , o — tt, tts — r= ^ 

,, „2 , ^ 2x2-1- 3x+ 1' 2x2- 17x-f 21' 

and 

4 x* - 20 X -f- 25 

Process. 

x2-6x-7 4x«-l 



4x=»-l 


4 x2 - 20 X -f- 25 


4x«-16x-|-15 


2 x« -h 3 X -H 1 


(2x-3)(2x-5) 



"2xa-17x+ 21 ^4x2-20x-f25 

(x-7)(x-hl) (2x+l)(2x-l) 
(2x-|- l)(x-|- 1) ^ (2x-3)(x-7) ^ (2x-5)(2x-5) 

(2x - 3) (2x - 5) (x - 7) (X -H) (2x -f 1 ) (2x-l) _ 2x-l 
(2x-|- l)(x-f l)(2x- 3)(x-7)(2x-5)(2x-5)~ 2x-6' 



182 ELEMENTS OF ALGEBRA. 

Explanation. Factoring the iiiuneiators and denominators of the 
fractions, multiplying the numerators together for the numerator of 
the product, and the denominators together for the denominator of 
the product, we have the third expression. Reducing the third ex- 
pression to its lowest terms, gives the result. 

Notes : 3. Observe the importance of factoring the terms of the fractions 
first. Also, indicate the multiplication of the numerators and denominators, 
and divide both terms of the fraction by their H. C. F. before performing the 
multiplication. 

4. If the factors are mixed expressions, sometimes it is better to change 
them to fractional forms before performing the multiplication. Thus, 

/ ah \ / _ ah \ _ a^ iA _ a^lfi 

V^ a-b)\ a + b)~a-b a + b~ a^-b^' 

2 x^ -\- 3 X 4 x^ Qx 

Example 3. Find the product of — r—^ — and lo^ + is ' 

Process. 

2a;2 + 3a; 4a:2_6a:_a;(2a; + 3) 2x(2x -3) 
4x8 >< 12a;+ 18~ 4x^ ^ 6 (2 a; + 3) 

_ x(2a: + 3) X 2 a: (2 a; -3) _ 2a:-3 

~ 4 a;8 X 6 (2 a: + 3) ~ 12 x * 

Exercise 69. 

Simplify : 

^ a2 j2 c2 3a3 2h^ 7c^ ^ ^ 

^- Fc^'^c^ aV 4.c^ "^ 21 a^"" 5 ah' ^ r' 

Sah^ 3«c2 Sad^ Sc-^x^ 2() (^ x 

2. 



4crf 2hd "^ 9hc ' 5a^y-^ 9a-^y-^ 

x+1 x + 2 x-1 Za^-x 10^ 

^' ^^^=~l^ x^-1^ {x + 2f' 5 ^2a;2-4ic 



a^ + 3 .r 4- 2 x^^-'Jx-\-\2 ^ m^ - n^ m^n 
^ _^ 9 a; + 20 ^ :i-2 _|_ 5 ^ + 6 ' ^3 _ ^2^ ^3^ ^£ 



X' 



6_ 



ALGEBRAIC FRACTIONS. 183 



2/6 ^■\-'f x + y 



X 



^- a^ + 2 3^ ij^ + 1/ s^ - xy -{- 1/ "" a^ - f 

am fm (i\ m^ 4- w* ^i f _!!!: ^ \ 

* oTw \a~7wy' m^ + n^ \m—n m + nj 

m^-\-mn nfi — n^ a^—(a-^b)x-{-ab x^—<? 
"' m^'^z ^mn{m-\-ny a^-{a + c)x+ ac^ x^-l^' 



m° — 



m^ + n^ f.. ^ ^ 

^' m^ — m n + n^ m^ + mn + n^ \ m — nj 

9. g _ ? + 1 V^2 + ^ + 1) • Suggestion. 

[e-)-i][(s-)-g-(s-)"-0'- 

^^ fx a y h\ (x a V h\ 

10. ( + ?--Ix( i + -]' 
\a X b yj \a X b yj 

\bc ac ab a J \ a + b + cj 

a^ •\- ab — ac (a + c)^ — V^ ab — b^ — be 

a:2--2a;"- 63 ^ a^- + 3a:"-40 ^ ?M^4^^+3 ' 
r a^> y'"* 1>.r (^'^-?/^"')^ 1 



8L Example 1. Find the quotient of -. divided by ^' 

fl c 

Solution. Let x represent the quotient. Then t -^ -5 = «. 
Since the quotient multiplied by the divisor gives the dividend, 



184 ELEMENTS OF ALGEBRA. 



we have x X -. = j^. Multiplying both members of the equation 

d c d a d ad 

by - , we have a:X-;X- = TX-, oraj^yX-* 
•^c' d c c^ he 

Therefore, ^^^=:-^X-=^. Hence, m general, 

To Divide a Fraction by a Fraction, invert the divisor, and 
proceed as in multiplication. 

Notes: 1. Since an entire or mixed expression may be written in fractional 
form, the above method is api)licable to all cases. Thus, 

_^a _ c _^a _ c ^__^c a _ a c _a 1 a 

^ ' b"! ' l~l^a~ ~^'' b'^^~l^\~b c~bc' 

2. It is usually better to change mixed expressions to fractional form before 
performing the division. Thus, 

(_a6\^/ ab \ _ <fi ^ b^ _ a'^ a + b __a^ 

" T+b) ■ V aT6/ ~ a + 6 ' aTl> ~ ^TTb ^ ~W~ ~ P ' 

^. ., ar2-14a:-15, x^-l2x-45 
Examples. Dmde ^,_^^^^^ by ^^--^-_^. 

Process. 

a:g-14ar-15 . g;2-12a;-45 (x - 15) (x + 1) , (x - 15) (a: + 3) 
a;2-4a;-45 ' a;2-6a:-27 ~ (x-9)(x + 5) ~ (ar-9)(a; + 3) 

- (a^-15)(a:+l) (re -9) 
~ (a: - 9) (a: + 5) ^ (x - 15) 
_ (a^-15)Ca;+l)(a:-9) x+ 1 
" (a: - 9) (a: + 5) (a; - 15) ~ a; + 5 ' 

^. ., a?^ 1 , a: 1 1 
Example 3. Divide ^+ibyp--+-- 

Process. 

\y^ x) ~ \y^~ y x) ~ xy^ ' x y^ 

_ {x -\- y) (x^ - X y + y^) xy^ 

xy^ x^— xy + y^ 

_x + y __x 



ALGEBRAIC FRACTIONS. 186 

Exercise 70. 
Divide : 

2 a^ 2:1 y , a x^ y"^ Z m , 2 m 

6 (g 6 - ^/2) 2 62 2;3_y3 (a; - y)8 

•^- a (a +6)2 '^^aCa^-feZ)' a^ -^ f ^ {x + y^' 

x^-f x-y ^ a^^xy+ip' ^ - 1?' 

m8+8^m + 2' 2? ^- f ^ 3?-xy-\-y^' 

^ 2^:2+13^^15 22:2_^ 11^ + 5 

5- ^^_Q by 



4a:2_9 -^ 43^2 



a:2 4- a: y + ?y2 x + y m 



m 



6. ;;o ^^-H^ by — ^^; -^ ^ by - + -. 

x^ — xy-)r]rx — y n^ m^ 71 m 

yj X-^y x-y x+if x-y X y 
' X ^ y X + y ^ X — y' x -\- y ^ y x' 

x^-^ (a + c)x + ac x^ - o? 

^- a?» + (6 + ^) a: + 6 c ^ ar» - 62 • 

a2 4.^>2_^2ft6~c2 « + 6 + c 
^- c2-a2_i,2+2a6 ^ 6 + c-a' 

10. a;8-^ by a:--; a2-62-c2+2&c by ^^44^^- 
a:^ -^ a: -^ a + 6+c 

11- ^6 . ^6 by -2-7-0; -e— r by 



7i6 + a:« ' 7i2 + a:2» fl^6__i ^ a^^a^^a-l 

^^ x~^ — x~^. x~i -{- x~^ 

12. 0^-8 by 



2a:-8 4a^2(x-f-a:-*) 



186 ELEMENTS OF ALGEBRA. 



Exercise 71. 

Perform the operations indicated in the following and reduce the 
results to their simplest forms : 

7 ic + 6 . a:2 + 6a; \ . . a;2 + lOa; + 24 

48* 



/x^-7x + 6^ x'^ + 6x\ x^ + 10a; + 
^* \x^-h 3a; -4 ~ x^ - 8 x^J ^ a;^- 14a; + 

/ x2 - 7 a; + 6 a;2 + 6a;\ a;'^ + 10 a; + 24 
Process. (^^2 + 3 a; _ 4 "^ a;3-8a;2j >^ a;^- 14a; + 48 

-i [ (^ - 6) (a; - 1) ^ a;(a; + 6) 1 (a; + 4) (a; + 6) 
~ [(a; + 4) (a; - 1) "^ x'^ {x - 8) J ^ (a; - 6) (x - 8) 



_ f a; -6 a;(x-8) 1 (a; + 4) (a: + 6) 
~ [a; + 4 ^ a; + 6 J ^ (a; - 6) (x - 8) 

_ a; (x - 6) (a; - 8) (x + 4) (a; -f 6) _ 
- (x + 4) (X + 6) (X - 6) (x - 8) ~ '^' 



a-l a + 1 «2_i 

X : 

a + i a — l'a + 



1' Va-&~ a + &J * a2-62- 

\x + y X — y x^ — y^J \x -\- y or — y^J 

^2 _ ^ _ 20 x^-x-2 x^-S6 ^ + 1 
4. — 5 :t?— X -0-7-r. o X 



x^-2d x^ + 2x-S^x^-6x ' x^ '\- 5x 

?/4 a + h x^—3xy + 2y^ ^ {^ — vf 

X 7 ; ZTt X 



' a^h + ah {x -\- y)^ x^ + y'^ ' ah 

/a-f Z) « — &\ /« + & a -- &\ 

max a^ — x^ h c -}- h x c — x ^ mx 

noy c^ — x^ a^ + a X a — x ny 



ALGEBRAIC FRACTIONS. 187 

8 1 • PM I Mx ^"^ 1. 

' x-^y \_2\x + y x — y) x^y + xy^J 

x^-\-x-2 aP-^5x+4: . f x^+Sx-{-2 x+3 \ 

^^' \6x-62 * x^^l^J'^ax+a^' 8a^y^^ 21b"'^^y'^-^ 
X 6^mH^-a^ (a: - 2)^ , ar^ - 4 
^a' ,^2-4 ^ 8 ?w?i + a * (ir + 2)a* 

12. ., o a X — =— i- X '^ 



14. (a:* — -jjH-fa; J, by inspection. 

15- (p - 2 + ^2) - (? - I)' ^y inspection. 

16. ic^— -g — sfa; j \-^ix ), by inspection. 

6flg^>g , r 3a(m-7i) . 5 4(c-a;) ^ c^ - a^ ^ 
m+n * I7{c + x) ' I 21afe2 * ^(m^-n^ij 



188 ELEMENTS OF ALGEBRA. 

82. A Complex Fraction is one having a fraction in its 
numerator or denominator, or both ; as, 

n' , m * 
n + — 
n 



Example 1. Eeduce - to its simplest form. 

c 

d 

Solution. A complex fraction may be regarded as representing 

the quotient of the numerator divided by the denominator. Hence, 

a 

h a ^ c a d _ a d 

cj^l^d'^'b^'i^'bc' 

d 

h 
a — - 

Example 2. Reduce f to its simplest form. 

m 

Solution. Since the divisor is m, we have 
6 

\c — b m ac — h 1 ac — h 



= [a — ~\ -^ m = - 



x- = 



I c m cm 

I 
I m , m 



Example 3. Reduce ~ » T » and — to their simplest forms. 



^ 1 m n n 

Process. — =zi-f- = i x — = — 

m n mm 



m 1 n 

Y = wi-r- = m X ^ = mn. 

n 

1 

m 1 1 1 n n 

-r-=~"^~ = ~ X7 = -. Hence, m general. 

I m n m I m ' & 



ALGEBRAIC FRACTIONS. 189 

To Simplify a Complex Fraction. Divide the numerator by 

the denominator. 

Example 4. SimpUfy "*' " ^^' ~ '"' + '^^ 



m + n m — n 



m — n m + n 
Process. 



m -j- n m — n ~ (m -h ?i)2 — (771 — n)^ ~ 4 wm 



m — n m 4- n (m — n) (m + n) (m — n) (m + n) 

4 m^n* (m — n) (m + n) in n 



X 



(m« - n2) (m^ + n^) '^ 4 mn m* + n^ 



Example 5. Simplify ^ ^ iiJ2 



x-y x+y 

Solution. Multiplying both terms by (x — y) (x + y), the L.C.D. 
of their denominators, we have 



2x1/ 



Notes : 1 . In many examples it is advisable to multiply both terms of the 
fraction by the L. C. D. of its denominators at once. 

2. If the terms of the complex fraction are complicated, the beginner is 
advised to simplify each separately. 



mp 



Example 6. Simplify ^'+(^+^03:+ mn x^ + {m+p)x + mp 

X* + (n + /?) X + n p 



190 ELEMENTS OF ALGEBRA. 



Process. 



mp mn mp 



v'^+(m-^n)x + m7i x^-h{m+p)x + mp _ {x-\rm){x + n) (x+m)(x+p) 
n — p ~~ n ~- p 

x'^+ {n+p)x + np {x + n){x+p) 

m n {x-\-p) — mp (x+n) 
_ {x-\-7ri) {x+n) {x+p) _ inx (n ~- p) {x + n) (x+p) 

n — p ~ {x + m) {x+n) {x+p) n—p 

(x + n) {x+p) 
rax 



X + m 



Example 7. Simplify T+~x' 

1 '- 



l-x + x'^ + 



a:2_ 1 

1 + X- 



X 



Solution. Begin with the complex fraction x^—\ ' '^^"®» 

a:2 _ 1 x+l , X x'^ 

i + x-—— = -^, and ^5^n[ = rri- ^'"^"^^^^y 

i + ^-^r- 

l +x^ (l + x^){l + x) ,^ l+x^ 
• ^= ,:8 + ^2+i ,and 1-- x^ 

l-X + X^+ ——r l-X + X^ + 






x^ + x^+l 
Therefore 



' 1 +x^ X 

1 X x^ + x^+l 

1 - X + X^ + -o 7 



1 +x- ——- 

X 
= - (X8 + X2 + 1). 

Notes : 3. A fraction of the form in Example 7 is called a Continued 
Fraction. 

4. To simplify a continued fraction, the student should always begin with 
the last complex fraction in the denominator. 



ALGEBRAIC FRACTIONS. 191 

Exercise 72. 
Keduce to their simplest forms : 

a; + 6 + ^ 1 + - a: + - xy 

^ j: — 6 wt c m n 



1 ' 6 ^ ' mm n ^ 

X-6 + 1 a;+- + 271 

X + o m n X 

, . 6^ 1 . 1 ni b m + n 

a-\-b + ^ - + — . 

^ a n m n m 4:mn 

n mm n S m^n 



a;+la;-l ar8-17a;+72 



2 x-1 ./J + 1 , 2:2 ^ 22 a; 4- 120 
^^ + 1 x-V x" - 21 a: +708 
x^^ iiTTl a?» + 18 a; + 80 



1 + « 4a 6 \a^; aj\x a) 

l + a6 2a6 a; + a 



J, + J_.JL 



- mn mp np f x , 1— A_/ x \—x\ 

' m^-(/i + pF ^ Vh^ ~^y \i+^"""^/ 



6. 



m 71 
1 1 1 



^ 1 ' 1 ' ^-1 • 

«.' + J 1 X — 

x + - 1 + i ^ + — ^- 

a; a; a; — 1 



192 

7. 



ELEMENTS OF ALGEBRA. 
1 X-2 



1 + 



l + a; + 



'1^ 
1-x 



x-2 



x-1 



x + 



y 



X 



X 1 , 1' _L 
y^ y X 



x — 2 

2 xy 

~ (^- + yf 



1+ 



x+ 1 



d + 



m 



1 + 



^,y 



(^' - yf X 



ax 
a? «2 






X' 



X 



X 



X x^ 



ir 



+ -0 -0+-+1 



a a? 



2/^ ^ 



X — y x" y^ 
y ^ 



10. 



11. 



w? + n^ 2 m 
m^ — ii^ ' 7?< + n 


~m n — m^ ^ m -{• n 
(m — iif' ' 'IV — n_ 



m — n 



71 + 



m — n 
1 + m n 



{in — n) n 



1 -\- mn 



m 


m 


— n 


1 - 


• m n 


1 - 


(m — 


n)m 



1 — mn ^ 



fm n\ 
\n m) 



83. Example. Find the third power of t; 



Solution. Since an exponent shows how many times an expres- 
sion is taken as a factor, we have 



'aJ^\ 



{aJ^y 



J^nj - 6" >^ &« >< ftn - (hny ~ h^n ' 



Hence, 



To Find any Power of a Fraction. Raise both terms of the 
fraction to the recjuirecl power. 



ALGEBRAIC FRACTIONS. 



193 



Exercise 73. 
Expand, by inspection, the following: 

^(-•-^)' m' [-G)'^a)T 

/ 2aHixi y' f ix + ;/f y r m (x - y) !^ 

r (r+,,)(x-,,) y r («-i)-»(«-5)n « /a^a-5)i\w 
*• L "« + » J' L (2«+3t)i J' ^ ^ J 



«-C4?)U©-©T= 



+ 1 



arff X 



wi 



ai 



Jx 



7«^ 



84. Example 1. Find the rth root of 



6" 



Solution. Since the rth power is found by takinp^ the numerator 
and denominator r times as a factor, the rth root is found by taking 
the rth root of each of its terms. The operation is indicated by 
dividing the exponent of each term by r. Thus, 

13 



194 



ELEMENTS OF ALGEBRA. 



5«4- 



Illustration, y 243^ =^ ssTs^^isTs - 3-p • ^®^c®» 



To Find any Root of a Fraction. Take the required root of 
each of its terms. 

Example 2. Find the square root of^2-"2 — aa;+j + — + ^- 



Arranging according to powers of a, we have 



X a 



Process. 

First term of the root squared, 

First remainder, 

First trial divisor, a^ 

a 
First complete divisor, « + - 



- times first complete divisor, 



Second remainder, 
Second trial divisor, 



4 + ^ + ^^-"^-^ + ^^ 



x^ a^ 



a2 + 



2a 



X 

2 a X 
Second complete divisor, a'^ + -— — - 



— - times second complete divisor, 



a8 


+ 


a2 

X^ 








X 




















rc2 








-ax- 


-2 + 


a2 



-aa;-2 + 



a2 



Note. If we take — |- for the square root of ^ , we shall arrive at the 



result — -p: f- -- . 

2 a; a 



ALGEBRAIC FRACTIONS. 195 

Exercise 74. 

Find the values of the following expressions : 

1- Vy^286' V a^ ^ V 343 ar^* ^ ^243 ^.25" j • 

f ni^n^ \^ ( Z2a^\\ ( (SA.m^n^x^ \\ 
\ a^^ ) ' \ b'^ ) ' \ 12oaH^7/y ' 

Find the square roots of : 



^, .. ^^ . 4 . ^^^ . ^ 






Miscellaneous Exercise 75. 

Reduce to lowest terms : 

h(b-ax) + a(a + bx) 2^-9 a^ + 7ix^ + 9 x-S 
(b-axf-h{a + bxf' a^ + Ta:^- 9 ^ - 7a;+ 8 * 

21 x^ y^ - S5 i/^z- 12 x^z + 20 xyz^ 
l8x^z^-21x^y-S0y2^+S5xfz' 

40r ^y*-?>2 2^ yz^ -5i/^z^ + 4xs^ 
40:^28- 36a:*22'-5/2 + 4oa^2/8* 



196 ELEMENTS OF ALGEBRA. 

^3~6a;2-37;r + 210 ^ a;4»+10 ^""^ 35 rg2"+ 50 a;^+ 24 
■ 0^+4x^-^7 X-2W ic3»+9^2n_|.26a:" + 24 

Find the values of : 

-_^2a;2/^2;^, . - ^a — ic 
0. o x^ -] o when x = 4, v = -k, z = 1: r: 

when 2; = 



a + b 



„ x^ + y''^ — z^ + 2xy . ^ . 

^ 2; — a ,x' — 6 , ft2 X X 

7. — ^^ when x = 7; - + 



b a a—b a b—a a+b 

a^ (b — a) 
when X = , ' , — ^ - 
b (b + a) 

, ^ ... ^ — 2 a + b a + b 

8- < ^ - + a-2b ^^^'^ ^ ^ ~2~ • 



(X — «V x 
X — b ) X 



9. i7^[_r + V ^ 2 J when x=^, ?/ = 1. 



2ab-\-2bc^-2ccl-^2ad ' y(a_c)(J+c)+6(-c-&)(a-2&) 
when « = 3, 5 = 1, c = - 2, t? = 6. 

g^ + g c 4- &2 ^ {4^ai^yij^ ^^\ c 

^2 — « c + 52 V'4a& — 52-2^ ~ a + 6 + cH-fZ 
when a = 4, & = 3, c = 1, c? = 7. 

Divide : 

12. !»J-^by™-x;^+gby?+2^. 



ALGEBRAIC FRACTIONS. 197 

1 , \ t:^ m^ . X m 
13. m* i by i?i ; -4 r by - H 

14 a^ + ^byx + -; a;3 ^ ^ + ^ _ _ by ^ - - . 
15. a2_j2_,2_26cby ^44^'- 



16 



• ^+i-K^^-^0-''(^'^-^)'^^-'^ 



Factor : 

Simplify : 

20. 3x-{y + [2..-(y-.)]} + i + |^^. 

a;— 1 x—\ a; + 3 a: + 3 

"3"*^^^ . 7 ~^T4 

^■^- a; + 2 a; 4- 2 * a;— 2 x—2' 

'~T'^x^ 3 '^0^1 

a-\ l-\ c-\ 

3a5c a h c 

' be -h ac — ab la.1 1 

a 6 c 



198 ELEMENTS OF ALGEBRA. 

23 ^ , ^ I ^^ + ^ , 



9^>2-(4c-2 a)2 Uc^-{2a-uy^ 4.a^-{U-4.cf 
' (2a+3&)2-16c2 + (36 + 4c)2-4ti2 + (2a+4c)'^-9&2 

\in/' j\m—n J \n^ J \m^-\-m7i+n^ J 

h 

a + 



1 X -{- a 1 X — a i_L^ 
27. ^IZ+f! + 5^Z±Z; ^ X («^-5«). 

1 a + a; 1 a — x' h ^ ' 

a 0^ + Q(^ ^ <x2 _}_ ^2 ^ 

h 

a — h — c h — c — a c — a — h 

a^—ac — ab + bc b'^—ab — cb+ac c^—bc—ac+ab 



X X 

11 1 



30. 



2/ + r 



ALGEBRAIC FRACTIONS. 199 



31. 



/ 3 2: + 3:3 X2 
yi-f 32:V 



9^ _ 33 - ar^ 3 



^ 32:2+1 ahc 



a:3_3a:'^ 2:^ (a^-a:)^ hc^ ac ab 



S-a-b + c 62 a-16-2 a^b-^ (a-^b-^y 



a ^b — c * a^ ab~^ 



aH~^ / a-2 6-^ Y 



'^2:'"+?/" '^2^^'"+?/2'* a-U^ &Ki ai(^ 
2:-+y" x^+y^ jic-i 

10 2; 7/ -3^2+ 10a:-3y _^ lQ2:-3y _3_ 
15i/^+102;//2 + :30y+202:y ' 452/ + 302:y ■*" y+2 ' 

- (f-)(i^r')H"-')(Sa?-) 

35 ^ + ^'' ^ + ?^^\.f + y') ■ \ff x) 

(x + y)^ — xy 
(x — yf-\-xy 

r g<-?/ ^ rt2+a?/ 1 To^-oV g^-2qg.v+fl27/n 



37. * 1 



Sax — 5b y ax '^by'^ 

1 ^""3a2;-2fey 

38. -i-^+-l-+ 1 



1 + i— 1 +-T— 1 + 



-t z X -\- z X -{■ y 



200 ELEMENTS OF ALGEBRA. 

39. (a + 2> + .)^- + ^ + -j- ^^^ 

(c - a) {a-h)'^ {a- h) {h - c)'^ {h - c) (c - a)' 

fa^—ij^ lOx^—lSxy—Sy^ 2a:^ + xy^+xy+y^\ 

\x^—jf l<)oiP'—Zxy—y^ 2x^ + xy—%f' ) 
xy — y'^— 2x+2y 
~ 2a;-?/ 

(a + hf - 6-3 {h + cf - g g (ci + c)3 - 63 



44. 
45. 



(a + i) — c h -\- c — a a ■\- c — h 

11 1 



a(a — &)(a — <?) b(b — a)(b — c) abc 

2a + n a + b + 71 m + n—a 



am+ab—bm—o? ab+bm—am—b^ m^—bm—am+ah 



46. -7 rc^ X + TT, TT^ -X + 



a(a—b)(a — c) b(b — a){b — c) c{c—a)(c—b) 

Queries. Why does changing the sign of one factor of either term 
of a fraction change the sign of that term ? Will it change the sign 
of the fraction ? Why 1 When the denominators have no common 
factors why multiply both terms by the product of the denominators 
of all the other fractions 1 Why does the process of reducing to 
forms having a common denominator not change the value of a frac- 
tion ? How prove the methods for addition, subtraction, multiplica- 
tion, and division of fractions 1 



FRACTIONAL EQUATIONS. 201 



CHAPTER XV. 
FRACTIONAL EQUATIONS. 

85. Example 1. Solve -j^ - 147^315 = "gT 30~ 

J_ 
■*"105* 

Solatioii. Multiplying each member by 210 (the L. C. M. of 
15, 21, 30, and 105), transposing and uniting like terms, we have 

— j = 5 + 30 X. Multiplying each member of this equa- 
tion by X— 1, transposing and uniting like terms, we have 25a; = 100. 
.-. x = 4. 

Proof. Substituting 4 for x in the given equation, we have 

6-5X4 7-2X4" _ 1+3X4 _ 10 X 4 - 11 1 
15 "14(4-1) ~ 21 30 "^105* 

or, — ^^ = — ^yV' which is an identity. Hence, 

To Clear an Equation of Fractions. Multiply each member 
by the L. C. M. of the denominators. 

o c, 1 2x+U 2fx-l x-l 

Example 2. Solve =— ^ - ^ = -^ • 

5 50 z — 10 2^ 

Proceas. Multiply by 5, 2 x + 1^ - loT^ = 2 x - 1. 

Transpose and unite, - r^ — — ^ = — 2^. 

Clear of fractions, - (2^ a: - 1) = - 25 x + 5. 

Transpose and unite, 22.6 x = 4. .*. x = ^W* 



202 ELEMENTS OF ALGEBRA. 

Note. In solving a fractional equation, where some of the denominators 
are simple and some are compound expressions, it is better to multiply each 
member of the equation by an expression which will remove the simple denom- 
inators tirst, then transpose (if necessary) and unite like terms. Similarly 
remove the compound denominators of the resulting equation. 



Exercise 76. 

Solve the following equations : 

■ X ^ Vlx~ 24' 2 '6x ~ ^ ~ 'Ix ' 

6a;+13 _ 32:4-5 2x 2^-5 x-?> _4a:-3 ^ 
15 5x-25 ~ 5 ' 5 "^ 2i^=^ ~ 10 i^- 

9 2^+5 8^-7_36^+15 lOJ 
14 ^6^+2" 56 "^l4" 

4a:+3 73;-29 _ Rrr+19 3,x+2 _ 2a:-l ^ a; 
9 "^5:^-12" 18 . ' 6 3a:-7 2* 

^ \^x-Tl ^ I+I62; ,. 101-642; 
^- ^9^^6^+2^ + -^4-^^^^-— 24 



6. 



18 2^+10 72 a; + 30 _ 20.5 16 2; - 14 
42 168 ~ 42 18aj+ 6 



1 2 _2;+2 4(2;4-3) _ 82;+37 72;-29 
^- 2"^'2:+2~ 22; ' 9 "" 18 52;- 12 

^ 2 2: + 8r} _ 13 2; - 2 , ^ _ 7^ _ 3^+16 ^ 



9 17 2^-32 ' 3~ 12 36 

g+ 1 22:-4 22;- 1 x-2 x-4: 
T5~ 72;-16~ 5 ' .05 -0625 



FRACTIONAL EQUATIONS. 203 

86. Frequently it is better to unite some of the terms before 
clearing the equation of fractions. Thus, 

X 

^^ ~ 3 16x4- 4.2 23 

Example 1. Solve -^^y + g^.^^ = ^ + ^Tl * 

X 

^^~3 23 16X+4.2 ^ 
Process. Transpose, — ^ - ^q-[ + -3^+2" = ^- 

,, . ,, ^~3 I6X+4.2 ^ 

Unite like terms, jTTj" H — g , « = 5. 

Free from fractions, 4+V-x-a:^+16x2+20.2x+4.2 = 15a:^25a:+10. 

1.6x 
Transpose and unite, — g— = 1.8. 

.-. a; = 3f. 

Example 2. Solve -^—^ - jqjg " ^a^Ti = 0- 

Process. Multiply by x^ - 4, (x + 2) - (x - 2) - (x + 1) = 0. 
Simplify, -x+3 = 0. .-. x = 3. 

Notes : 1. If a fraction is preceded by the — sign, in clearing the eqiiation of 
fractions, care must be taken to change the sign of each term of the numerator. 
In such case it is convenient to enclose the numerator in parentheses before 
clearing the equation of fractions. 

2. The student should be careful to observe that he can make but two 
classes of changes upon an equation without destroying the equality : 

I. Such as do not affect the value of the members. 

II. Such as affect both members equally. 

Thus, in the above process, the first operation affects both members equally; 
and the second, that of uniting like terms, does not affect the value of the 
members. 

4 2 5 24 

Example 3. Solve ^-j^ - ^^^^ = ^-^ - ^-^ . 

Solution. Transposing, ^ - g^ = ^ - 2F+2 ' 



204 ELEMENTS OF ALGEBRA. 

Simplifying each member separately, we have 
3 _ 11 1 



2 (x + 3) 2 (a: + 1) ' "^ a; + 3 - 2 (a: + 1) 
Clearing of fractions, we have 2 (a; + 1) = x + 3. r. x=l, 

-c^ A oi ^ — 4 x — 5 x — 7 x — 8 

Example 4. Solve r 

x-5 x-Q x-S x-9 

Solution. Reduce the fractions to mixed expressions, 

1 1 1 1 ^ , . , 

or ^-3-^ - ^^^^ = ^-jg - ^^^ • Reducmg the terms in each 
member separately to common denominators and adding, we get 

- (a:-5)(a;-6) =" ~ (a: - 8) (a; - 9) ' ^^^^""^ *^^« ^^"^^^^^ «f 
fractions, we have —{x — 8) (^ — 9) = — (x — 6) (x — 6). Simplify- 
ing, transposing, and uniting like terms, — 6 a: = — 42. .'. x = 7. 

(2 a; + 3) a; J^ 
2a;+ 1 "^ 3a; 

dx A- 3^ X 

Process. Reduce ^^ , i to a mixed expression, 

Transpose and unite, - ^j:^ == - 3^ ' 

Clear of fractions, — 3a; = — 2a:- 

Therefore, a; =.- 1. 

-n. r. , 5a;-64 2a:-ll 4a;-55 

Example 6. Solve 



a:- 13 x-Q x-14: x-1 

Process. Reduce the fractions to mixed expressions, 



FRACTIONAL EQUATIONS. 205 



Simplify each member separately, 

7 



(x - 13) (x - 6) ~ (2: - 14) (x - 7) 
Divide by 7 and clear of fractions, 

x2 - 21 a; + 98 = a?2 - 19x + 78. 
Therefore, x = 10. 

Exercise 77, 
Solve the following equations : 

12 1 29 x + 4 x+6 



a; ' 12a; 2.4* 32;-8 3a:-7 

3a:+l a;-2 6a:+l 2 a; - 4 2a;-l 



3(a:-2) a;-l' 15 7a:-16 5 

x-\-25 ^ 2x + 75 5 4 _ 3 

a;-5 "" 2a:-15' 1 - 5 a: "*" 2 a; - 1 "" 3a: - 1 



6a;+8 2a;+38 _ x^^x+1 a^+x-{-l _ ^ 
2a:+l a;+12 ~ * a;-l "^ a: + 1 " ^^' 

^7_2^--15 1 J 2 1__ 

a:+7 2x'-6"*"2a;+14"~ '1-a: 1+a: l-x^~^' 

3 30 3.5 



4 - 2 X- 8 (1 - a;) 2 - ./; 2 - 2 a; 

6^-7i l + 16a: , 121-80. 

^' 13-12a:+'^^+ 24 ^ ^^^ 3 

a;— 1 a; — 5 a; — 4a;--2 
aj — 2 a; — 6 a;— 5 a* — 3 

5a;-8 6a;-r44 10a;-8a;~8 
' a;-2 ar — 7 a; — 1 "" a; — 6 ' 



206 ELEMENTS OF ALGEBRA. 



x-l x+1 _ 2(x^ + 4x-tl) 
^^ • x-2'^ x + 2~ {x+2f 

, , 30 + 6 rr 60 + 8 a^ ^ . 48 

11. ; — H ; — rt — = 14 + 



X + I X + O X + 1 

.6a:+.044 .5^--.178 _ .3a^-l _ .5 + 1.2 a; 

■^^' A .6 ^-^^^ .5x-A~ 2x^1 

2x-Z Ax -.6 1-lAx _ .7{x-l) 



13. 



.3ic-.4 .06;:c-.07' x + .2 .1 - .b x 



87. A Literal Equation is one in which some known 
number is represented by a letter; as. 



X X 

Example 1. Solve — f- 



m n — m m -jr n 

Process. Clear of fractions, x {n"^ —m^)+x (m^ +mn) = m^{n- m) 

Unite like terms, {n^ +mn)x = m^{n — m). 

m^(n—m) 
Divide hy n(n + m), x = ^^^^^_^^^y 

Example 2. Solve (x-m) (x-n) — (x-n) {x-a} = 2(x-m) (m-a). 

Process. Simplify, transpose, and unite, 

Sax — 3mx=: — 2m^+ 2am ~ mn -\- an. 

Factor, 3 (a --m)x = (a — ?«) (2m + n). 

2m + n 
Divide by 3 (a — m), x = ^ — • 

a2-3&x ,„ , 6 6a:-5rt2 

Examples. Solve ax ab^ = ox-\ -^ 

a 2a 

bx + 4a 

4 

Process. Clear of fractions, simplify, transpose, and unite, 

4a^x-3abx = 4a^b^- 10 a^. 
Factor, a (4 a - 3b) x = 2 a^ (2b^ - 5). 

Divide by a (4 a - 3 6), • x= ^\_.^f^ ' 



FRACTIONAL EQUATIONS. 207 

_, ax ~b bx — a a — b 

Example 4. Solve , . — l^ , ,. = /„ ^ , h\ /k ^ ^ „\ ' 

ax -{■ ox + a {ax + o) {ox + a) 

Solution. Reducing the terms of the first member to mixed ex- 

/ 2b \ f 2a \ a-b 

pressions, we have [l - -^-^ j - [l- ^^^j = (^^^^^^^^^^^ • 

Uniting like terms and reducing the fractions to a common 

denominator, adding and factoring their- numerators, we have 

2(a-{-b) {a-b)x a-b ^,, . ^ , . 

7 . iv /I. — ; — 7 = 7 , .V .. — ; — ; . Clearing of fractions, 

{ax + b) {bx + a) {ax + b) {bx + a) " * 

2{a + b) {a — b) X = a — b. Therefore, x — ^ . . v • 

Notes: 1. Example 4 may be solved by clearing tlie equation of fractions. 
The solution is presented as an expeditious method. 

2. If the student cannot readily discover a special artifice, be should clear 
the equation of fractions at once. 

3. Known terms are called absolute terms. Thus, in the equation mx^ 
•\- nz -\- a = 0, a is called the absolute term. 

a -i- b a b 

Example 5. Solve ; = 0. 

x — c X — a X — b 

Process. Clear of fractions, 

{a-\-b){x-a){x-b)-a{x-b){x-c)-b{x-a)(x-c) = 0. 

Simplify, transpose, and factor, 

x{ac + bc - a^-b^) = ab{2c -a-b). 

Tx. ., , , « .„ ab{2c-a-b) 

Divide by a c + 6 c - a* - 62, x = 7^; 5 — A* 

^ ' ac + bc — a^—b^ 

_fl &(a + 6-2c) 

°' ^~a2 + 63-c(a + 6)' 

Exercise 78. 

Solve the following equations : 

10. e a 6 1 

X a ^x 

2. \0hmx — ^an = 2am — hhnx\ = r* 

a X X 



208 ELEMENTS OF ALGEBRA. 

^ 7n? n 4:71^ m a h „ ,« 

X A X 4: ox ax 

4.-|.(.-«)-(^-±^J=^(.-|). 

5. ^^ - ^^ +2 = 0; (x-a)(x-h) = {x'-a-hf. 

^ a{b^x + a^) ax^ 2fx A Zfx \ 

6. -^— V ~ aca; + ,-; -- + 1)=-- — 1). 

hx b S\a ) 4\a / 

3 ah — x^ 4:X — a c x^ — a a — x 2 x a 



^ ^ ^ ^ ^^ w^.^ l^ ^ ^ 




' c hx ex ' hx h 


h 


^ X — m 0? — mx — V? ^ n? 

L — 1 • 




VI mx — n^ mx — n^ 





Miscellaneous Exercise 79. 

Solve the equations : 

X 07+1 ^ — 2x'^ac hc_ , 

9 ^ ~1> 1 - 9aj' h~x~'^~^ '^ ' 

ax+h Sh ^ a^x^ + h^ 

ax — h a X + h~ a^x^ — b^' 

X X ■\- \ _x — ^ X — ^ 

' x — 2 X — 1 X — ^ X — 1 ' 

2(2a;+3) 6 bx+\ 



4 1 



63-9^ 1-x 2% -Ax 



FRACTIONAL EQUATIONS. 209 

5 1 I 1 1 0. 

a (6 — a;) b{c — x) a(c ^ x) 

6. (2a:--^)rar+^^ =4a:^^-a:Vj(a--4a;)(2a + 3a:). 

17 __ . _ 105 +10a; _ .^ 
^* a; H- 3 ^ ~ 3 a: + 9 

8. (.^3)^_^^^) = 7.-(3.-?i^)). 

a?— a a + a; 2aa;_ 1 ^ 1 __ a — h 

' a—b a -{• b a^—l^~ * x — a a;— 6 x^—ab 

10. 3— + — j — = 2 a;; -=c(a — ^)) + -. 

a; — la; 41 x ^ ^ x 

_ a:+ 2 , a:- 7 a: + 3 x - ^ 

11. h ■= — — — z- = 7 • 

a; X — o X + 1 X — 4: 

135 a; - .225 .36 .09 x - .18 



12. .15 a; + 
13. 



.6 ~ .2 .9 

x—a x—a—1 x^b x—b—1 



a;-a — 1 a? — a — 2 x—b — l x—b — 2 



^ . SO a — bx 9 n — ax 6 m — nx 
14. = 5 ^ = 0. 

,^ 4m(a2-5./2) ^ 5 m (J^ - 2 a;) 

8a: 4 



X — np X — mp X — mn 

16. p = p. 

^ m n V 

14 



210 ELEMENTS OF ALGEBRA. 



3 & (a; — a) a; — &2 ^ & (4 a + c a?) 
5 a 15 6 ~ 6a 



mx — n mx -\- n 

c^ — Sdx(P+2cx X X 

' c^+odx d^—2cx~~* m ~~ n 

n ',n 



m x 7n(x—m) x(x + m) mx 
x m x{x + m) m{x—m) m^—x^ 



Queries. Upon what principle is an equation cleared of fractions ? 
How is it done '? Why change the signs of the terms of the numera- 
tor of a fraction, preceded by a minus sign, when clearing of fractions '? 
Upon what principle (give four different explanations) may the signs 
of all the terms of an equation be changed ? 



Exercise 80. 

1. The second digit of a number exceeds the first by 3 ; 
and if the number, increased by 36, be divided by the 
sum of its digits, the quotient is 10. Find the number. 

Solution. Let x — the digit in tens' place. 

Then a; + 3 = the digit in units' place, 

and 2 a; + 3 = the sum of the digits. 

Therefore, 10 a: + a: + 3, or 11 a: + 3 = the number. 

lla:+3 + 36 
Hence, — ^ — r-5 — = 10. .-. a:= 1. lla:+ 3 =14, the number. 



PROBLEMS. 211 

2. The first digit of a number is three times the second ; 
and if the number, increased by 3, be divided by the differ- 
ence of the digits, the quotient is 17. Find the number. 

3. The first digit of a number exceeds the second by 4 ; 
and if the number be divided oy the sum of its digits, the 
quotient is 7. Find the number. 

4 The second digit of a number exceeds the first by 3 ; 
and if the number, diminished by 9, be divided by the 
sum of its digits, the quotient is 3. Find the number. 

5. A can do a piece of work in 7 days, and B can do it 
in 5 days. How long will it take A and B together to do 
the work ? 

Solution. Let x = the numler of days it will take A and B to- 
gether. 

Then - = the part they do in one day ; 

but = = the part A can do in one day, 

and e = the part B can do in one day. 

Therefore, = + ^ = the part A and B can do in one day. 
7 o 

Hence, - = ^ + ^. Therefore, x = 2\^. 

6. A can do a piece of work in 2 J days, B in 3 days, 
and C in 5 days. In what time will they do it. all work- 
ing together ? 

7. A can do a piece of work in a days, B in 6 days, 
C in c days. In what time will they do it, all working 
together ? 



212 ELEMENTS OF ALGEBRA. 

8. A and B together can do a piece of work in 12 days, 
A and C in 15 days, B and C in 20 days. In what time 
can they do it, all working together ? 

9. A and B together can do a piece of work in a days, 
A and C in 6 days, B and C in c days. In what time can 
they do it, all working together ? In what time can each 
do it alone ? 

10. A tank can be emptied by three pipes in 80 min- 
utes, 200 minutes, and 5 hours, respectively. In what 
time will it be emptied if all three are running together ? 

11. A sets out and travels at the rate of 9 miles in 5 
hours. Six hours afterwards, B sets out from the same 
place and travels in the same direction, at the rate of 11 
miles in 6 hours. In how many hours will he overtake A ? 

Solution. Let x — the number of hours B travels. 

Then x + 6 = the numher of hours A travels; 

also, I = the numher of miles per hour A travels, 

and i^- = the number of miles per hour B travels. 

Then, y^ x = the number of miles B travels, 

and I (a: + 6) =r the number of miles A travels. 

Hence, V" ^ = f (^ + 6). Therefore, x = 324. 

12. A man walked to the top of a mountain at the rate 
of 2 miles an hour, and down the same way at the rate of 
3^ miles an hour, and is out 13 hours. How far is it to 
the top of the mountain ? 

13. A person has a hours at his disposal. How far 
may he ride in a coach which travels b miles an hour, so 
as to return home in time, if he can walk at the rate of c 
miles an hour ? 



PROBLEMS. 213 

14. In going a certain distance, a train travelling 55 
miles an hour takes 3 hours less than one travelling 45 
miles an hour. Find the distance. 



15. The distance between London and Edinburgh is 
360 miles. One traveller starts from London and travels 
at the rate of 5 miles an hour ; another starts at the same 
time from Edinburgh, and travels at the rate of 7 miles an 
hour. How far from London will they meet ? 

16. The distance between A and B is 154 miles. One 
traveller starts from A and travels at the rate of 3 miles 
in 2 hours ; another starts at the same time from B, and 
travels at the rate of 5 miles in 4 hours. How long and 
how far did each travel before they met ? 

17. The distance between A and B is a miles. One 
traveller starts from A and travels at the rate ot 711 miles 
in n hours ; another starts at the same time from B, and 
travels at the rate of b miles in c hours. How long and 
how far did each travel before they met? 

1 8. If it takes m pieces of one kind of money to make 
a dollar, and ?i pieces of another kind to make a dollar, 
how many pieces of each kind will it take to make one 
dollar containing c pieces ? 

19. The denominator of a certain fraction exceeds the 
numerator by 6 ; and if 8 be added to the denominator, 
the value of the fraction is J. Find the fraction. 



20. A can do a piece of work in 2 m days, B and A 
gether in n days, and A and C in m + ^ 
time will they do it, all working together ? 



together in n days, and A and C in m + ^ days. In what 



214 ELEMENTS OF ALGEBRA. 

21. In a composition of a certain number of pounds of 
gunpowder the nitre was 10 pounds more than ^ of the 
whole, the sulphur was 4^ pounds less than J of the whole, 
and the charcoal 2 pounds less than ^ of the nitre. Find 
the number of pounds in the gunpowder. 

22. A broker invests | of a certain sum in 5 % bonds, 
and the remainder in 6 bonds; his annual income is 
$180. Find the amount in each kind of bond, and the 
sum. 

23. A broker invests — th of a certain sum in a % bonds, 

n 

and the remainder in c % bonds ; his annual income is b 
dollars. Find the amount in each kind of bond, and the 
sum invested. 

24. At the same time that the west-bound train going 
at the rate of 33 miles an hour passed A, the east-bound 
train going at the rate of 21 miles an hour passed B ; they 
collided 18 miles beyond the midway station from A. 
How far is A from B ? 

25. A person setting out on a journey drove at the rate 
of a miles an hour to the nearest railway station, distant h 
miles from his home. On arriving at the station he found 
that the train had left c hours before. At what rate should 
he have driven in order to reach the station just in time 
for the train ? 

26. A merchant drew every year, upon the money he 
had in business, the sum of a dollars for expenses. His 
profits each year were the nth. part of what remained after 
this deduction, but at the end 3 years he found his money 
exhausted. How many dollars had he in the beginning ? 



SIMULTANEOUS SIMPLE EQUATIONS. 215 

CHAPTER XVL 

SIMULTANEOUS SIMPLE EQUATIONS. 

88. Simultaneous Equations are such as are satisfied by 
the same values of the unknown numbers. 

Thus, 3 X + y = 9 and 5a: — 2y = 4 are satisfied only hy x = 2 
and y = S. 

Elimination is the process of combining simultaneous 
equations so as to cause one or more of the unknown 
numbers to disappear. 

This process enables us to fonn an equation containing but one 
unknown number. The equation thus formed can be solved as 
shown in the preceding chapter. 

Hote. There are only three methods of elimination most commonly used. 

Elimination by Addition or SnbtractioiL 

89. Example 1. Solve the equations : 5 3a; -5?/ =13 (1) 

^ l2x + 7y = S\ (2) 

Hote 1. The abbreviations (1), (2), (3), etc., read "equation one," "equa- 
tion two," etc., are used for convenience to distinguish one equation from 
another. 

Solution. To eliminate x we must make its coefl&cients equal in 
both equations. Multiplying the members of (1) by 2, and those 
of (2) by 3, we have 

5 6 X - 10 y = 26 (3) 
i6a; + 21y = 243 (4) 



216 ELEMENTS OF ALGEBRA. 

Subtracting the members of (3) from the correspojiding members 
of (4), we have 31 y = 217. .'.y = 1. Substituting this value of y 
in (1), we obtain 3 a; - 35 = 13. .-. x= 16. 

VerifiGation. Substituting 16 for x, and 7 for ^ in (1) and (2), 



we have 548-35 = 13 (1), 
we nave "[gg, 49^31 (2), 



, , identities. 
32 + 49 = 81 (2), 

Votes : 2. In this sohition we eliminate x by subtraction. But suppose we 
wish to eliminate y instead of x. Multiply (1) by 7, and (2) by 5, then add 
the resulting equations, and we have 31 ic = 496. . •. ic = 16. This value of x 
substituted in (1) gives y = 1. 

3. When one of the unknown numbers has been found, we may use any one 
of the equations to complete the solution, but it is more convenient to use the 
one in which the number is less involved. 

4. It is usually convenient to eliminate the unknown number which has the 
smaller coefficients in the two equations. If the coefficients are prime to each 
other, take each one as the multiplier of the other equation. If they are not 
prime, find their L. C. M., divide their L. C. M. by the coefficient in each equa- 
tion, and the quotient will be the smallest multiplier for that equation. 

Example 2. Solve the equations : 515^ + ^7^ = 92 (1) 

^ ( 55 a: - 33 7/ = 22 (2) 

Solution. Multiplying the members of (1) by 11 (the quotient 
of 165 divided by 15), and those of (2) by 3, we have 

5 165 a; + 847 y = 1012 (3) 

Xl^bx- 99 2/= 66 (4) 

Subtract the members of (4) fron the corresponding members of 
(3), 9461/ = 946. .-. ?/= 1. Substitute this value of y in (1), 
15ar+77 = 92. .-. a; = 1. 

Proof. Substituting 1 for x, and 1 for ?/ in (1) and (2), we have 

5 15 + 77 = 92 (1) 
1 55 - 33 = 22 (2) 

Hence, both equations are satisfied for a: = 1 and 2/ = !• 

Example 3. Solve the equations : 5 ^7 a: - 12 !/ = 289 (1) 

^ ( 55 a; + 27 2/ = 491 (2) 



SIMULTANEOUS SIMPLE EQUATIONS. 217 

Process. Multiply (1) by 9, 693 x - 108 y = 2601 (3) 

Multiply (2) by 4, 220 x+lOSy= 1964 (4) 

Add (3) aiid (4), 913 x = 4565. .-. x = 5. 

Substitute this value of x in (2), 275 + 27 y = 491. .♦. y = 8. 

Prool Substitute 5 for x, and 8 for y in (1) and (2), and we 
have J 279 = 279 (1), .^^^^^ 
\ 491 = 491 (2), 
Let the student supply the method from the solutions. 



Exercise 81. 

Solve the following simultaueous simple equations 



1. 


|3a; + 4y=10. 
Ux"+ y= 9. 


8. 


(Jy + Ja; = 26.» 
(fy + |.; = 25. 


2. 


Sx- y = 34. 
a; + 8 y = 53. 


9. 


( .25 x + 4.5y = 10. 
1. 75 y-. 15 a; = .9. 


3. 
4 


10 a: + 9y = 290. 
12 2;-lly = 130. 

7 y - 3 a; = 139. 
2x + 5ij= 91. 


10. 


J 3^ 2 '' 
l2 + 3 = ^- 


5. 


{6x-5y = -7. 
\ 10 a; 4- 3 7/ = 11. 


11. 


( .5 a: + 2y= 1.8. 
1 .5 y - .8 a: = .08. 


6. 


9a;-4y = -4. 
15 aj + 8 y = - 3. 


12. 


(7a:+^y = 99. 
1 7 y 4- j a: = 51. 


7. 


9 y 4-2 a; =15. 
4y4-7a;= 3. 


13. 


r Jaj4- 3y = 22. 
1 l\x-\y=20. 



♦ Clear of fractions first. 



218 ELEMENTS OF ALGEBRA. 



Elimination by Substitution. 

90. Example. Solve the equations : HaJ + 32/ = 22 (1) 

^ l5x-7y= 6 (2) 

Solution. From (2), x = — — -^ (3). Since the equations 

o 

are simultaneous, x means the same thing in both, the substitution 
of this value of x in (1), will not destroy the equality. Hence, 

4/ — F~^) +3^ = 22. Clearing of fractions, transposing, and 

uniting like terms, 43 2^ = 86. .'. y = 2. Substitute this value of 
y in (3), x = 4. 

Let the student supply the method. 



Exercise 82. 

Solve by substitution : 

1 



2. 



3. 



x+Sij^U. ^- l^x + iy = 7. 

7 x + 4:y = 29. {S7J + 4cX = SS. 

Sx+ ?/ = ll. \5x+62j=61. 

l^V-^x = 21. l3+2 = l- 



I .08 2/ - .21 a = .33. ( 3 y - 4 a; = 1. 

I .7z + .12y= 3.54 I 3 a; - 2 ?/ = 1. 

" ~ *• 10. I 11 -^ 

^-- = 0. 




^ = 1. 



SIMULTANEOUS SIMPLE EQUATIONS. 219 

Ml2/-7^ = 37. (10a: = 9 + 7y. 

^^- |8y + 9a: = 41. U2/ = 15a!-7. 

12. < aud verify. 

Elimination by Comparison. 

91. This method depends upon the following axiom : 

6. Things equal to the same thing are equal to each 
other. 

Example. Solve the equation . )^^-^y=^ 0) 

^ (7x-4y = 8^ (2) 

Solution. From (1), x = i±A^ (3). From (2), x = ?i±l^. 

Since these equations are simultaneous, x means the same thing 

in both, — ^ = -^ ^ . Solving for y, we have y = 4. Sub- 

7 1 + 20 

stituting this value in (3), x = — tj — = 3^. 

Let the student supply the method. 

Exercise 83. 
Solve by comparison : 

5a;+6y = — 8. j6x+l5y = — 6. 

3x + 4y = -5. 



(6x+ l5y = -{ 
I 8 a; - 21 y = 74 

\}x-{-iy = S. 

(-^x + 3y = 51. rSy -.7x =.4. 

' \7x+2y = 3. I .02 y + .05 a: = .2$ 



12^-7^=17. 
• U2:+8y = 20. ^ 



220 ELEMENTS OF ALGEBRA. 



(l.lx -l.Sy =0. 
^- t.l3:r- .11 7/ = .48. \l + 



0^ = 3 2/ -23. ^^1 5 2 ~'^* 



12. -{ ^ 



1 = 42. 



r.30^-.772/=:-2.95. h^ 4.y^4o 

1 .20 a: +.21^=1.65. ^8"^ 9 



92. Each of the equations should be reduced to its simplest form, 
if necessary, before applying either method of elimination. 

Notes : 1. An expeditious method, for the solution o^ particular examples, 
is that of first adding the given equations, or subtracting one from the other. 

2. Usually, in solving examples of two unknown numbers, it is expedient 
to find the value of the second by substitution; but this is by no means 
always so. 

Example. Solve : 

2y + 42:-2tf 10|i/-5fx-18 

3a:+.y 13?/-37^_ 9-9a;-t/ 10a:+.25i/-10.5 

L~T2^+ 44 -^^+ 22~^~ 33 ^^^ 

Process. From (1), 127 y+ 59 x= 1928 (3) 

From (2), 59 ^ + 127 rr = 1792 (4) 

Adding (3) and (4), 186 y + 186 a; = 3720 (5) 

Dividing (5) by 186, y+ a; = 20 (6) 

Subtracting (4) from (3), 68?/- 68 a; = 126 (7) 

Dividing (7) by 68, y - x = '2. (8) 

Adding (6) and (8), %y =22. .-.2^=11. 

Subtracting (8) from (6), 2x~ 18. .-. a:= 9. 



Solve 



SIMULTANEOUS SIMPLE EQUATIONS. 221 

Exercise 84. 



1 fy(^ + 7) = a:(y+ 1). (2y + Ax =1.2. 

I 2y+20 = 32;+ 1. XsAy -.02x= .01. 

r(y+l)(2:+2)-(y + 2)(2:+l) = -l. 
^- \ 3 (y + 3) - 4 (2: + 4j = - 8. 

f .3 2: + .125y = a:- G. rx-4y = -3. 

l3a:-.5y = 28~.25y. I aj + v =32. 



.5y = 28~.25y. {x + y 

6. -^ 



'4:X + Sy 2 y -\-7--x _ X--S 
To 24 -^"^ 5 



9a;+52/ — 8 a; + y _ 7y 4- 6 



12 

4 



9. ^ 



<^ 5 + y 12 + a; 
l2a;+ 53^ = 35. 




10. -^ 



3y-10(a:~l ) ^j-y . . _ ^ 
6 -^ 4 "^ ^ - "• 

' 4a;-3y-7 _ 3_^ _ 2 .y _ 5 
5 " 10 15 6 * 

y~l , ? _ 3y _ 7/^^ 4. ? 4. JL 
L 3 "^2 20 15 "^ 6 ■*■ 10 



-3 5 ~ 4 * U"^il 33 



222 ELEMENTS OF ALGEBRA. 

r:. + l(3^.-y-l)^i+f(2/-l). 

f2x _Sy — 2 _ _ 4: + X y — x 



14. ^ 18 

by 



\2x. 

2Zy -X 



2x-^ = 



a? + 43 X — y 
2.4 X + .32 y 



\ X -]r "i^o X — y 

.36 X - .05 



\'. 


1-3.V 

7 


= 2i. 


3^/ + 
11 


-^-9 = 


= — ii;. 


« ^ _L 


2.6 + 


.005 3/ 



17. 



.5 ' .25 

04 2/ + .1 .07 a? -.1 



18. 



.6 



_ Zx-2-\-,y ^ IBy + jx 

^ 11 "^33 



2x-h3y x-5 11 :?/ + 152 3 a; + 1 



r 



?/ - 2 10 - y ^ - 10 



19 ^ ^ ^ ^ 

■^22; + 4 0^ + 4?/ +12 



I 



8 




21 H(2^+ 72/)- 1=1(20.-62/+!). 

I a? =: 4 2/. 



SIMULTANEOUS SIMPLE EQUATIONS. 223 

Suggestion. Multiply the members of the first equation by 2, 
transpose, and unite like teims ; then clear the resulting equation of 
fractions. Multiply the second equation by 3, transpose, and unite 
like terms; etc. 



23. < 



24. < 



25. 



r2 , y ^ 3?/ 1 

,1-1 + 2 = 1-2.. 6. 

f 6x + 9 3x-5 _ 3x + 4: 

4 +4y~6~ i"^ 2 
8a; 4- 7 Sx- 6y _ 9-4a ; 
^10 2a:-8~ 5 

16 + 60x _ 16xy- 107 
3y-l ~ 5 + 2y • 



Suggestion. Multiply the members of the first equation by 
5 + 2 y, transpose, and unite like terms ; then clear of fractions ; etc. 



26. 



27. 



X — y __1 
X -\- y 5 * 

13 3 



y+2a;+3 4?/ — 5a;+6 

3 ^ 19 

6y-5a; + 4~ 3y + 2a;+ l' 

ry-x = l. r5(y+3) = 3(ic~2) + 2. 

28. I y+ 1 _ y-1 _ 6 29. ^ 2 ^ 3 
U— 1 X ~ 7' i^y+3~a: — 2* 



224 ELEMENTS OF ALGEBRA. 



30. 



31. 



32. < 



i(2 2/ + 7^) -1 = 1(2 2/- 6^+1). 



U = 



y 




6y2-24a^+130 
2y-4:x+ 3 
151 - 16y _ 9 0^3/ -110 
4^-1 3a;-4 ' 

^4a; + 22/ 4a;+53/ 
""16 31 "" • 

2^+j/ 3 ?/ - 2 0? 36 

~5 +""~6 =y 

r5.T + 202/ = .l. 04 J 2/-^ 3-"- 

^''- \ll^ + 302/ = -.9. l^+_^±i_7^Q 

V2/ — a; — 1 



35. 



r 2a;— .5y 5jy— 19a;— 15 „ ^— a; + 2 



93. Fractional simultaneous equations in which the unknown 
numbers occur in the denominators as simple or like expressions^ are 
readily solved without previously clearing of fractions. Thus, 



Example 1. Solve: 



^'h 


21 

y 


= 


10 


20 


6 






X 


y 


- 


2 



(1) 

(2) 



Solution. 

10 3 
Dividing the members of (2) by 2, we have — — - = 1 (3). Mul- 

70 21 
tiplying the members of (3) by 7, — — -— = 7 (4). Adding the 

X y 



SIMULTANEOUS SIMPLE EQUATIONS. 225 

members of (1) and the corresponding members of (4), we have 

. X = 5. Substituting this 



85 5 

— = 17. Dividing by 17, - = 1 

X X 



value of X in (1), gives - = 1: .. y = 3. 

Note- If we cleared these equations of fractions they would give the pro- 
• liict xy, and thus become quite complex. In the solution of this particular 
.lass of examples it is always easier to eliminate one of the xmknown numbers 
without clearing of fractions. 



Example 2. Solve: 



2-y^rx = ^ 
2 4 _ 



2 20 136 
Process. Multiply (0 by i, 3T + g^- = ^ 



Subtract (3) from (2), 
Simplify (4), 



y 

4_ 
5x 



2,1 x 
20 
■ 27 X 

8 
135 X 



208 

9 
208 

9 • 



2 2 

Substitute in (2), ^ - 312 = -8, or « = 304. 

^y ^y 



(1) 

(2) 

(3) 
(4) 



^~ 390' 



y = 



456 



Exercise 86. 



Solve: 



1. < 



2. 



r2 1 ,^ 
- + - = 10. 

X y 



9 y 2x~ ^' 
^3y 4 a; 6 * 




226 



ELEMENTS OF ALGEBRA. 






5_ 
12 

24 



12. 



?-^ = 16. 

2/ 2a; 

14,^ 
+ - = _ 15. 

l2y X 



7. 




y + l 



__7^ 
~ 12' 

12* 

_ 5 

~ 6' 

= 2. 



13. 



14. 




2 2/ 4a? 



13^/^22; 



71 
'1- 



79. 



15. < 



5 16 

- + — 
X y 

= 44. 

y^x y 



9. 



ri5 


8 


17 


2/ 


X ~ 


3 


2 


3 


7 


I y 


a; 


5 



16. 



11 

2^ 


+ 


6 
32/ 


17. 


17 

6 1/ 


— 


5 
a; 


3 
2* 



10. 



1 



+ 



2 (2; - 2) ' 3 (2 2/ 
3 5 



1) 



5 a; -10 4 (4 2/ -2) 



= 5. 



= 1. 



11. 



22/ 2 



17. 



2a; 



2 


5 


4 

"27 


1 

42/ 


1 


11 

"72 



18. 



SIMULTANEOUS SIMPLE EQUATIONS. 227 

a; — 2 y + 2 ^g 

3 1 1 



^a; — 2 y +• 2 2 



1 1 7 
2x 3y""15* 
a: — 5 v/ + 4 a: y 


1 


4 a; V 


15 



19. Suggestion. Reduce the first member of the second equa- 
tion to mixed expressions. Etc. 

('2i _ 3y ^ 2 rL2. 25^86 

l3^ 3y-2a;+l 3'2lJ^2^ 
5, 2y _ * |25_1_6^ 

a;"^3y-2a;+l la; y 



94. In solving literal simultaneous equations, either of the pre- 
ceding methods of elimination may be applied, usually the method 
by addition or subtraction is to be preferred. 

Note. Numbers occupying like relations in the same problem, are generally 
represented by the same letter distinguished by different subscript figures ; as, 
«l ; «2 ; "8 > fitc* ? r«*d a one ; a ttoo ; a three ; etc. 

They may also be represented by different euxents ; as, a'; a"; a'"; etc.; 
read a prime; a second; a third; etc. 



Kt AMPLE 1. Solve: 1^ ^ 


+ n 


y 


= a 


(1) 


-hn, 


iV 


= «i 


(2) 


Process. Multiply (1) by 


mj. 




m^mx + m^ny = m^a 


(3) 


Multiply (2) by m. 






m^mx + mn^y = ma 


(4) 


Subtract (4) from (3), 






m^ny — mn^y = mjO - 


-w»«i, 


or factoring, 






(m^n — mn^)y — m^a - 


- may 


Dividing by mjn — muj, 






^""h 


- Oj m 


^ nij^n - 


-mnj 


Multiply (1) by rij. 






mn^x -f- nn^y = n^a 


(6) 


Multiply (2) by n, 






m^ux -\- nn^y = na^ 


(6) 


Subtract (6) from (5), 






mn^x — m^ux = 71^0 - 


-no,. 


or factoring, 






(mui — in^n)x = n^a - 


- noj. 


Therefore, 








-_a^n 
-m^n' 



228 



ELEMENTS OF ALGEBRA. 



Example 2. Solve: ^ 



f X y 

1^+5 + ^='" 



x + 



(1) 

(2) 



[ 2a6 ~ a2 + 62 

Proceas. Free (2) from fractions, transpose, and factor, 

{a-hyx-{a^-hyy^Q (3) 

Simplify (1), {a-h)x^{a^lS)y = 2a(a+b) (a-b) (4) 

Multiply (4) by a -ft, (a-byx+(a^-b'^)y = 2a(a+b) {a-by (5) 
Subtract (3) from (5), 2 a (a + 6) i/ = 2 a (a + 6) (a - &)2. 

Divide by 2 a (a + 6), y = (a-b)\ 



Substitute in (1), 



Examples. Solve 



a+b 



+ a-b = 2a. .'.x=(a + by. 



f m n—'m(m+n)(b—y)__ 

J n{a-\-x) m(b-~y) ~ 

! ryt 



I — — + T = m2+n2 

\^a + x b—y 



Process. From (1), 
Multiply (3) by ~, 



+ 



n(a + x) m{b — y) 



= m + n 



+ 



a + X m^(b — y) m 



— (m4-n) 



(2) 

(3) 
(4) 



Subtract (4) from (2), ^3^ - ^2Q,_y^ 

1 



Simplifying, 
Substitute 



m{b — y) ^ m 



m(b-y) 



= 1 or 



b-y 

n 
a + x 



m2 in (2), 



= n^ .-. X 



Example 4. Solve: < 



x-y +\ 
x-y-1 

x + y + I 



x + y 



- a = 
-6 = 



(1) 
(2) 



SIMULTANEOUS SIMPLE EQUATIONS. 229 

Procesa. From (1), {a- l)x - (a- l)y = a + I (3) 

From (2), (6- 1):. + (6- l)y = 6 + 1 (4) 

Divide (3) by a - 1, x~y = ^—^^ (5) 

Divide (4) by 6 - 1, x + y = ^-3-j (6) 

2(a6-l) 
Add (5) and (6), 2 x = (q_i)(fc_i^ ' 

ab-l 
•••^-(a-l)(6-l)- 

2(a-6) 
Subtract (5) from (6), 2 y = ^^. ^^^^.^^ • 

a-b 



Exercise 86. 

Solve : 

- (ax-\-hy = m. ^ ( ax -\- by = a^, 
'\bx-\-ay = n. ' \hx -\- ay = 1?. 

^ nx + my = n. ^ (px'-qy = r, 
' \px-\-qy = r, ' \rx—py = q, 

^ (ax = hy. ^(x + ay = ai. 
' \bx-\-ay = c, ' \ax-\-aiy=l. 

-? + ?=!. (^ + ^- = a. 

. J a b ab ^J^ \ m n 

I bi aibi ^n ' m 



5. < 



3j/^2^^2 r_y X ^ 1 

m ?i * -^ Ja-{-b a — b a + b 

9y_6^^3 ' I ?/ , ^ ^ ^ 



230 ELEMENTS OF ALGEBRA. 

= (a + h) y. 



' \ cy + hx = a. ' \x + y = c. 



rih; / J 14. - + 2' = 2. 

l-^ + r = cll+-). {mx = ny 



\^(m — n) y = {m -{• n) X. 



y X ^ 
a ai 




(ax-hy 

\(a-h)x+ (a + b)y=2 {a^ - l^). 

22 (m{m-7j) = n(x + y-m). 
\m (x — n — y) = n (x — n). 

a b a fx + y+l^m + l 



2g^a + a? b-y b 24. <^2/-^+l ^"1 

b a b \ X + y + 1 __ n+1 

a + x b — y~a \y — x-^l~l—n 

25 / 3/ - ^ + 2 (m - ?i) = 0. 

■ 1 (a:; + 7i) (y + m) — (y — m) (x — n) = 2 (m — n)\ 



SIMULTANEOUS SIMPLE EQUATIONS. 



231 



95. Simultaneous equations with three or more unknown num- 
bers are solved by eliminating one of the unknown numbers from the 
given equations ; then a second from the resulting equations ; and so 
on, until finally there is but one equation with one unknown niunber. 
Thus, 

r 2 y + 2 + 2 y = - 23 (1) 

y-f-32 = - 2 (2) 

4a: + z=13 (3) 



Example 1. Solve 



3 + 3. 



Process. Multiply (2) by 2, 
Subtract (5) from (1), 
Multiply (4) by 12, 
Subtract (7) from (3), 
Multiply (8) by 5, 
Add (9) and (6), 
Substitute in (4), 
Substitute in (3), 
Substitute in (2), 



- 20 (4) 

2y + 6z = - 4 (5) 

-52 + 2y = - 19 (6) 

4a; + 361; = -240 (7) 

z-36v= 253 (8) 

52- 180i;= 1265 (9) 

-178y= 1246. .-. r = - 7. 

I -21 =-20. .-. x = 2. 

12 + 2=13. .-. 2=1. 

y + 3 = -2. .-. y = -5. 



Proof. Substituting — 7 for y, 3 for x 

f - 23 = - 

- 2 = - 
13=13 

- 20 = - : 



(I), (2), (3), and (4), we have -{ 



— 5 for y, and 1 for z in 
23 (1), 

^ ^^]' identities. 
(3). 

(4), 



Kote. When the values of several unknown numbers are to be found, it is 
necessary to have as many simultaneous equations as there are unknown 
numbers. 



EiLAMPLE 2. Solve: 



J_ J 1__ 1 

2z"^ 4y 32~ 4 



1 

1 4 



0) 

(2) 
(3) 



\2 ELEMENTS OF ALGEBRA. 






Process. Multiply (1) by 2, - -f 


1 2 1 
2y 3z-2 




(4) 


Subtract (2) from (4), 




5 2 1 
6y~ 3z~2 




(5) 


Subtract (2) from (3), 




2 4 
I5y'^ z~^^ 




(6) 


Multiply (5) by 6, 




5 4 




(7) 


Add (6) and (7), 




77 77 
16 y~ 15- 


.'. y = 


:1. 


Substitute in (2), 




1 1 


.'. X- 


:3. 


Substitute in (5), 


fl 1 


5 2 1 

6 3z~ 2' 


.'. z - 


:2. 
(1) 


Example 3. Solve : - 


1 1 

ly + i-" 

Process. 




s 


(2) 
(3) 



Add (1), (2), and (3), ?+?+^ = a + & + c (4) 

Divide (4) by 2, ^ + J + 1 = ^±A±f (5) 

Subtract (3) from (5), ^ = ^±|-— 

Subtract (2) from (5), ^ « + c - 6 



Subtract (1) from (5), 



y 2 

1 &4-C — a 





2 




«+&-c 




2 


y — 


a-6 + c 




2 



h-\rC — a 



SIMULTANEOUS SIMPLE EQUATIONS. 



233 



Exercise 87 

Solve: 

f'^x- y+ z= 9. 
I. ^ a:_2y+32= 14. 



r4:x-3y+2z = 40. 

^<5x-i-9y-7z = 4:7. 

Ua;+8y-32 = 97. 

r2x-32j+oz= 15. 

3. < 32:+27/- z= 8. 

V— a;+ 5y + 2s = 21. 

rSx-Sy+ z= 0. 

4.<2a;-7y + 42:= 0. 

v9 2:+5 7/+32= 28. 

rx -\- y -\- z= 5. 
6. ^ 3 7/-5x + 72 = 75. 
19 y- 11 2+ 10= 0. 

r.65//- .95a: = .5. 

6. < 5.1 a: -3.3 3 = 6. 

V20.3 2- 23.1a: = 21. 

rax •\- hy '\- cz=iZ. 

7. < rt x — 6 y + c 2 = 1. 

Vaa: + 6y — • C2 = 1. 

r.2a; + .ly + .32=14. 

8- < .52:+.4y+.a2 = 32. 

^.7y-.8a:+.9c= 18. 



9. < 



2^ + 2 + 3 = 

-+2+1= 
a; ?/ 



X y z 



^ + - + ' 



6. 

--1, 

17. 

= 1. 
= 1. 
= 1. 



11. < 




234 



ELEMENTS OF ALGEBRA. 



fv-hx + y + z = 14:. 
\2v + x = 2y + z-2. 
14^^ ^ Sv - X + 2 y + 2 z = 19. 
\ V X y z 



^a — X h — y c — z 

+ + = 0. 



X 



y 



15. < 



a — X h c _ 
X y z~ 

X y z 



0. 



15. Suggestion. Reduce fractions to mixed expressions. Etc. 



'x-\-2y = 9, 
16. ^3 7/ + 4^ =14. 

72 + V = 5. 

^2v + 52:== 8. 
rx + y= 1. 
\x -\- z = b. 



2 1_ 3 

X y z 

18. < ^ - - = 2. 

z y 

1 1_4 

x^ z~ ^' 



f4:y-\-Sx + z 2x + 2z-y-\-l _ y-z-5 



10 



15 



19. < 



9?/ + 52:-2;s 2y + a; — 3^_ 7x + z+S 1 

r2 ^ 4 ~ 11 ^ 6 * 
5^ + 32 2y+Sx-z ^ ^ Sy + 2x+7 
—4 12 + 2. = 0.-1+ g 



Queries. Upon what principle is elimination by addition and 
subtraction performed ? What substitution ? What comparison ? 



SIMULTANEOUS SIMPLE EQUATIONS. 



235 



Miscellaneous Exercise 88. 
Solve : 

'x+1 x-l 6 r4a;+y = ll. 




23 



y 



a(x^-y) + h{x-y) = l. 
& (x + y) = 1. 



7. 



4. 



5. 



10. 



X — a 



= 0. 



a b 



0. 



'a; + 2y=2-32-4i;. 

Sy + 2x = S — 4:z-5v. 

9v-82-3 = -6a;--7y. 
^v = 25 -4^- 16y- 64a:. 

■(m2 — n2) (5 ^^ 4. 3 y) = (4 ^ _ n) 2 m n. 

„ a m vF 
m^y 



3 a; 15 


/M- 


lf-!=»- 


Ihh' 




y ^ 

12-3- 


.|!+l-^ 




1=^ 
,ai bi ' 



-{-{m-\-n-\'a)nx = n^y-\-{m-\-2n)mn. 



11. 



3 V + a: + 2 y - 2 = 22. 
4a;- y4-32 = 35. 
4v + 3a;-2y = 19. 
.21^ + 4^+2^ = 46. 



(-15 a; = 24 2- 10 y + 41. 

12. \ 15y=12a;- I62+ 10. 

ll8y-(7 2-13)-=14a;. 



'Ul=z. 



13. < 



X 

1 

a; ■ z 



y 
2 
+ - = 11. 



^U-3. 
y 2 



236 



ELEMENTS OF ALGEBRA. 



14. 



'x + y + z + v = 14:. 
2x+Sy + 4:Z+5v = 54:. 
4:x — 5y — 7z+9v = 10. 
:Sx + 4:y + 2z-3v = ll. 



16. 



) X + Z + V = 



= 5. 
10. 

X + y+ V = 6. 
x + y + z=12. 



15. 



18. 



ax + by = 2m. 
ax -\- cz = 2 71. 
,h y + c z = 6p, 



rmx + ny + pz = m. 

17. < mx — ny—pz = n, 

\mx +py + nz = p. 



Sx 



2y , \ly 
— + 1+ ^ 



10 



45 



4.X-2 



8 7 

55^+ 71 y + 1 
18 



4 a; — 3 V + 5 45 ■ 

- + — 7 



^lx = 
v^2Z 



17 + 2z-Stc. 



20. 



2(0 + 22/). 
4. 

y= 2.25-\-.75u—5v. 
z = ll-^u. 



19. {u=ly^ I X 



( ax + 
\ay + 



b X 
by 



cy = m. 

cx = n. 



X y m 

21. < -+ - = -. 
\x z jp 

I 1 1_ 1 

\^z y n 



22. 



f :^+2/+2; = a+J+c. 
i a+x = b-\-y = c+z. 



9S 11-7^ 2(5-lly) ^ 17.5 + 5y 312.5-360a; 
?j-x'^ ll{y-l)~ 3-2/ 36(a; + 5) ~ 



^3 4 1 ^^ fx ^ ^ 

--^— + -= 7.6. -+l=4a;. 

X by z 

4 



_ .,- + - = 16.1. 

^o X 2y z 



+ l = 2y. 



SIMULTANEOUS SIMPLE EQUATIONS. 



237 



1 



y + 



1 



y~ 



26. < 



a; — 



X — 



i-i 



y 
l-x = 0. 



x-\-y = 2m^ 



27. 



711 -{-li- 



mn 



m+n 



m—n-\- 



mn 
m—n 



28. 



m n V ^ 
- + - + - = 3. 
X y z 

m n P _ ^ 
X y"^ z~ ' 



2 m 

X 



n 



0. 



29. 



30. ^ 



xy 

x-{- y 

xz 



= 70. 
= 84. 



X ■{- z 

-^ = 140. 

y + z 



31.^ 



xy 

x+ y 

yz 
y + z 

xz 
^x -\- z 



xy 

— ^^ = m. 



32. ^ 



'ax-\- by + cz = 0. 
a^x + ly^y + (^z = 0. 
.a^x + h^y'i-c?^z = 0. 

'a^z=:2. 



x + y 


xz 


x+ z 


yz _ 


^y + « 



n. 



{ax + a^y — a^yVc 

2(3a;-2y) _^_ bz-y 



34 



35. 



a; — 2g 
3a; — 2i 



m — n + 



Sz-7 

n 



2x — Sz 



m 






ly-^ = 



w 



m + 71 

2 7l6 



7/1^ + 7?l 71 + 71^ 



238 ELEMENTS OF ALGEBRA. 



CHAPTER XVII. 

PROBLEMS LEADING TO SIMULTANEOUS EQUATIONS. 

96. The solutions of the following problems lead to simultaneous 
simple equations of two or more unknown numbers. In the solution 
of such problems the conditions must be sufficient to give just as many 
equations as there are unknown numbers to be determined. 

Exercise 89. 

1. If 5 be added to both numerator and denominator 
of a fraction, its value is f ; and if 3 be subtracted from 
both numerator and denominator, its value is ^. Find the 
fraction. 

Suggestion. Let x = the numerator, 

and y — the denominator. 

ra; + 5_ 3 
— T~K — T' 

By the conditions, •{ y ^ 

I a:-3 _ 1 

Solving these equations, x — 1, y =\\. 
Therefore, the fraction is ^y- 

2. A certain fraction becomes equal to 3 when 9 is 
added to its numerator, and equal to 2 when 2 is sub- 
tracted from its denominator. Find the fraction. 

3. Find two fractions with numerators 5 and 3, respec- 
tively, whose sum is ||, and if their denominators are 
interchanged their sum is f. 



PROBLEMS. 239 

4. A certain fraction becomes equal to § when the 
denominator is increased by 3, and equal to J when the 
numerator is diminished by 3. Find the fraction. 

5. A fraction which is equal to | is increased to {| when 
a certain number is added to both its numerator and de- 
nominator, and is \ when 3 more than the same number 
is subtracted from each. Find the fraction. 

6. If a be added to the numerator of a certain fraction, 
its value is a ; and if a be added to its denominator, its 
value is ^ (a — 1). Find the fraction. 



7. Find two numbers, such that two times the greater 
added to one fifth the less is 36 ; three times the greater 
subtracted from eight times the less, and the remainder 
divided by 9, the quotient is 7|. 

8. Find two numbers, such that if the first be increased 
by a, it will be m times the second, and if the second be 
increased by 6, it will be n times the first. 

9. Find two numbers, such that if to J of the sum you 
add 18, the result will be 21 ; and if from | their differ- 
ence you subtract |, the remainder is 3.65. 

10. A farmer sold to one person 25 bushels of corn and 
52 bushels of oats for S 38.30 ; to another person 42 bush- 
els of com, and 37 bushels of oats for $35.80. Find the 
number of dollars per bushel received for each. 

11. A farmer sold a bushels of corn and b bushels of 
oats for m dollars ; also at the same time, c bushels of com 
and d bushels of oats for n dollars. Find the number of 
dollars per bushel received. Apply the result to 10. 



240 ELEMENTS OF ALGEBRA. 

12. A grocer bought a certain number of eggs, part at 
2 for 3 cents and the rest at 5 for 8 cents, paying $7.50 
for the whole. He sold them at 23f cents a dozen, and 
made $2 by the transaction. How many of each kind did 
he buy ? 

13. A grocer bought a certain number of eggs, part at 
the rate of a eggs for m cents and the rest at the rate of h 
eggs for n cents, and paid c dollars for the whole. He 
sold them at d cents a dozen, and made ^ dollars by the 
transaction. How many of each kind did he buy ? Apply 
the result to 12. 



14. A number is expressed by three digits. The sum of 
the digits is 8 ; the sum of the first and second exceeds 
the third by 4; and if 99 be added to the number, the 
digit in the units' and hundreds' place will be inter- 
changed, rind the numbers. 

Suggestion. Let 2 = the digit in units' place, 

and y = the digit in tens' place, 

also X — the digit in hundreds' place. 

Hence, 100a:+ 10y + 2 = the number, 
and 100 2+ 10^ + a: = the number with the digit in units' 

and hundreds' place interchanged. 
By the conditions, 
X + 2^ + z = 8, 
a: + 1/ — 4 = 2, 

100 a: + 10 y + 2 + 99 = 100 z + 10 2/ + a:. 
Solving these equations, z — % y = 5, x — 1. 
Therefore, the number is 152. 

Note 1. In verifying, the results should be tested directly by the conditions 
of the problem. Thus, in the above, the sum of 2, 5, and 1 is, as one condition 
requires, 8. The sum of 1 and 5 exceeds 2 by 4» The sum of 162 and 99 is 251 
fiB required. 



PROBLEMS. 241 

15. A number is expressed by three digits. The middle 
digit is twice the left hand digit, and one less than the 
right hand digit. If 297 be added to the number, the 
order of tlie digits will be reversed. Find the number. 

16. A number is expressed by three digits. The sum 
of the digits is 18 ; the number is equal to 99 times the 
sum of the first and third digits, and if 693 be subtracted 
from the number, the digit in the units' and hundreds' 
place will be interchanged. Find the number. 

17. The sum of the three digits of a number is n ; the 
number is equal to a times the sum of the first and third 
digits, and if m be subtracted from the number, the digit 
in the \mits' and hundreds' place will be interchanged. 
Find the number. 

18. If a certain number be divided by the sum of its 
two digits the quotient is 3, and the remainder 3 ; if the 
digits be interchanged, and the resulting number be di- 
vided by the sum of the digits, the quotient is 7, and the 
remainder 9. Find the number. 

19. If a certain number be divided by the sum of its 
two digits the quotient is a, and the remainder b ; if the 
digits be interchanged, and the resulting number be di- 
vided by the sum of the digits, the quotient is c, and the 
remainder m. Find the number. 

20. The sum of the three digits of a number is 16. If 
the number be divided by the sum of its hundreds' and 
units' digits the quotient is 77 and the remainder 6 ; and 
if it be divided hy the number expressed by its two right- 
hand digits, the quotient is 16 and the remainder 5. Find 
the number. 



242 ELEMENTS OF ALGEBRA. 

21. The sum of the three digits of a number is 9. If 
the number be divided by the difference of its hundreds' 
and units' digits, the quotient is 157, and the remainder 1; 
and if it be divided by the number expressed by its two 
right-hand digits, the quotient is 21. Find the number. 

22. A, B, and C can together do a piece of work in 12 
days ; A and B can together do it in 20 days ; B and C 
can together do it in 15 days. Find the time in which 
each can do the work. 

Suggestion. Let x = the number of days in. which A can do it, 

and y = the number of days in which B can do it, 

also 2 = the number of days in which C can do it. 

1111111 111 

Theequationsare- + ^ + ~ = 12' ^ + 2^= 20' ^^^ y + z = \b'^ 

from which a; = 60 and y = 2 = 30. 

23. A and B can do a piece of work together in 48 
days ; A and C in 30 days ; B and C in 26| days. How 
many days will it take each, and how many altogether, to 
doit? 

24. A and B can do a piece of work together in a days; 
but if A had worked m times as fast, and B n times as 
fast, they would have finished it in c days. How many 
days will it take each to do it ? 

25. A drawer will hold 24 arithmetics and 20 algebras; 
6 arithmetics and 14 algebras will fill half of it. How 
many of each will it hold ? 

26. A purse holds 19 crowns and 6 guineas ; 4 crowns 
and 5 guineas fill JJ of it. How many will it hold of 
each ? 



PROBLEMS. 243 

27. A purse holds c crowns and a guineas ; ci crowns 

tn 
and ai guineas will fill — th of it. How many will it bold 

of each ? 

28. A and B together could have completed a piece of 
work in 15 days, but after laboring together 6 days, A was 
left to finish it alone, which he did in 30 days. In how 
many days could each have performed the work alone ? 

29. Two persons, A and B, could finish a piece of work 
in m days; they worked together a days when B was 
called off and A finished it in n days. In how many days 
could each do it ? 

30. A can row 8 miles in 40 minutes down stream, and 
14 miles in 1 hour and 45 minutes against the stream. 
Find the number of miles per hour that the stream flows, 
also that A rows in still water. 

Suggestion. Let x = the number of miles per hour that A can 
row in still water, 
and y = the number of miles per hour that the 

stream flows. 
Then, x + y = the number of miles per hour that A can 

row down the stream, 
and X — y = the number of miles per hour that he can 

row up the stream. 
Since the distance divided by the rate will give the time, by the 
conditions, 

8 2 

x + y~ 3* 

31. A can row m miles in h hours down stream, and mi 
miles in ^i hours against the stream. Find the number of 
miles per hour that the stream flows, also that A rows in 
still water. Apply the result to problem 30. 



244 ELEMENTS OF ALGEBRA. 

32. A boatman sculls down a stream, which runs at the 
rate of 5 miles an hour, for a certain distance in 3 hours, 
and finds that it takes him 13 hours to return. Find the 
distance sculled down stream, and his rate of rowing in 
still water. 

33. A man who can row at the rate of 15 miles an hour 
down stream, finds that it takes 3 times as long to come 
up the stream as to go down. Find the number of miles 
per hour that the stream flows. 

34. A waterman rows 30 miles and back in 12 hours ; 
and he finds that he can row 3 miles against the stream 
in the same time as 5 miles with it. Fiud the number of 
hours in going and coming respectively ; also, the number 
of miles per hour of the stream. 

35. A waterman can row down stream a distance of m 
miles and back again in h hours ; and he finds that he can 
row h miles against the stream in the same time he rows 
a miles with it. Find the number of hours in going and 
coming, respectively ; also the number of miles per hour 
of the stream, and his rate of rowing in still water. 



36. Five pounds of sugar and 3 pounds of tea cost 
$2.05, but if the price of sugar was to rise 40 %, and the 
price of tea 20 % they would cost $2.51. Find the num- 
ber of cents in the cost of a pound of each. 

37. If / pounds of sugar and h pounds of tea cost m 
dollars, and the price of sugar was to rise a % , and the 
price of tea h %, they would cost n dollars. Find the num- 
ber of cents in the cost of a pound of each. 



PROBLEMS. 245 

38. The amount of a sum of money, at simple interest, 
for 11 months is S1055; and for 17 months it is S1085. 
Find the sum and the rate per cent of interest. 

39. The amount of a sum of money, at simple interest, 
for VI months is a dollars ; and for n months it is h dollars. 
Find the sum and the rate of interest. 

40. A grocer mixes three kinds of coffee. He can sell 
a mixture containing 2 pounds of the first kind, 9 pounds 
of the second, and 5 pounds of the third, at 18 cents per 
pound ; or one composed of 6 pounds of the first, 6 pounds 
of the second, and 9 pounds of the third, at 19 cents per 
pound ; or one composed of 5 pounds of tlie firet kind, 2 
pounds of the second, and 18 pounds of the third, at 22 
cents per pound. Find the number of cents in the cost of 
a pound of each kind. 

41. The fore-wheel of a carriage makes 6 revolutions 
more than the hind-wheel in going 120 yards ; if the cir- 
cumference of the foie wheel be increased by J of its pres- 
ent size, and the circumference of the hind-wheel by J of 
its present size, the will be changed to 4. Find the 
number of yards in the circumference of each wheel. 

42. The fore-wheel of a carriage makes a revolutions 
more than the hiud-wheel in going b feet. If the circum- 

ference of the fore-wheel be inci-eased by — th of itself, and 

8 ^ 

that of the hind-wheel by - th of itself, the hind-wheel 

r 

will make c revolutions more than the fore-wheel. Find 
the circumference of each wheeL 



246 ELEMENTS OF ALGEBRA. 

43. A grocer has two kinds of coffee. He sells a pounds 
of the first kind, and h pounds of the second, for m dollars; 
or, ax pounds of the first kind, and hi pounds of the second, 
for mi dollars. Find the number of dollars in the price of 
a pound of each kind. 

44. A jeweller has two silver cups, and for the two a 
single cover worth 90 cents. If he puts the cover upon 
the first cup it will be worth 1^ times as much as the 
other ; if he puts it upon the second cup it will be worth 
lyig times as much as the first. How many dollars in the 
value of each cup ? 

45. A jeweller has two silver cups, and for the two a 
single cover worth a dollars. If he puts the cover upon 
the first cup, it will be worth m times as much as the 
other ; if he puts it upon the second cup it will be worth 
n times as much as the first. How many dollars in the 
value of each cup ? 

46. A broker invests $5000 in 3's, $4000 in 4's, and 
has an income from both investments of $315.50. If his 
investment had been $1000 more in the 3's, and less in 
the 4's, his income would have been $5.50 greater. Find 
the market value of each class of bonds. 

Note 2. 3's means bonds which bear 3 % interest. The " quoted " price of 
a bond is its market value. Thus, a bond quoted at 115i means that a $100 
bond can be bought for $115.50 in the market. 

47. A broker invests m dollars in a's, n dollars in c's, 
and has an income from both investments of h dollars. If 
his investment had been d dollars less in the a's, and more 
in the c's, his income would have been p dollars less. Find 
the price paid for each kind of bonds. 



PROBLEMS. 247 

« 

48. A and B do a piece of work together in 30 days, 
for which they are to receive $1G0. But A is idle 8 days 
and B is idle 4 days, in consequence of which the work 
occupies 5J days more than it would otherwise have done. 
Find the number of dollars received by each. 

49. A and B do a piece of work together in m days, for 
which they are to receive c dollars. But A is idle a days 
and B is idle h days, in consequence of which the work 
occupies n days more than it would otherwise have done. 
Find the number of dollars received by each. 

50. The amount of a sum of money, at simple interest, 
for 5 years is S600; and for 8 years it is $660. Find the 
number of dollars in the sum, and the rate of interest. 

51. The amount of a sum of money, at simple interest, 
for a years is m dollars ; and for h years it is n dollars. 
Find the number of dollars in the sum, and the rate of 
interest. 

52. If a grocer sells a box of tea at 30 cts. a pound, he 
will make SI, but if he sells it at 22 cts. a pound, he will 
lose S3. Find the number of pounds in the box, and the 
number of cents in the cost of a pound. 

53. The smaller of two numbers divided by the larger 
is .21, with a remainder .04162. The greater divided by 
the smaller is 4, with .742 for a remainder. Find the 
numbers. 

54. The smaller of two numbers divided by the larger 
is a, with a remainder m. The greater divided by the 
smaller is h, with c ibr a remainder. Find the numbers. 



248 ELEMENTS OF ALGEBRA. 

CHAPTEK XVIII. 

EXPONENTS. 

97. An Exponent is a figure or term written at the right 
of and above a number or term (Art. 21). 

'^ m 

Thus, in the expressions 5^, a% 6", and (a + by; 2, c, —, and 3 

are exponents. 

Zero Exponents. When the dividend and divisor are 
equal the quotient is 1. 

Thus, ^,= 1; -,= 1; ~,= i; ~ = I; etc. 

But (Art. 30), 32 = 32-2 = 30; ^ = aO; ^^ = «^ ^ = «^ etc. 
Therefore, it follows that a^ = I. Hence, in general, 
I. Any expression with zero for an exponent is 1. 

The Reciprocal of a number is unity divided by that 
number. 

Thus, the reciprocal of n is -; of n + m is 



n -\- m 

Negative Integral Exponents. 





a3 X a-8=a8-3 = «o^i 


Divide by a^, 






a» X a-« = «"-« = a^= \ 


Divide by a", 


a~** = — . Hence, in 



EXPONENTS. 249 

II. A negative integral exponent indicates the reciprocal 
of the expression with a corrcspoiuling positive exponent. 

The expression a", where n is any positive integer, represents tht 
product o/n equal factors, each equal to a. It has been shown that : 

Art. 21, a^ X a* — a*»+". 

Art. 30, a"^ — a^ = a"*-", where m is greater than n. 

Art. 30, ti^ -T a'*= ^_,^ , where m is less than n. 

Art. 27, (a*")" = a*"", whatever the value of m. 

Thus, 

By Art. 21, a* X a" X a' x .. . . a"* = a»+«+ ''+••'». 

Take n factors of a*, a*, a**, a"*, and suppose each of the n ex- 
ponents equal to m, then it follows that 

(amy _ Qtnn^ Hencc, m can 
be positive or negative, i itegral or fractional. 

By II.. <'-" = ii- 

a"* 
Multiply by a"*, a-" X a"» = — • 

If m is greater than n, Art. 30, -;; = a*"-". 
Therefore, a-" >^ a*" = a"*-*. 



If m is less than n, 



a* 1 



a» a 



i« — w 



By II., 7=^ = «' 



a" 



Therefore, a-» x a*" = «"•-'• for all possible integral 

values of m and n. 



98. By Art. 27, (ai)" = a" X 6". 

Therefore, o-^*" = (aft)*. 

Similarly, a" X ft" X c" X . . . . p" = (a 6 c . . . . />)■ 



250 ELEMENTS OF ALGEBRA. 

If n is a negative integer, 



«-" X *-" = a« X 6" =(«6)» =("*)-• 






Similarly, 

a-« X i-^ X c-» X ....jD-"= {ahc ... 
general. 


. Py- 


Hence, in 



I. The product of tv:o or more factors, each affected with 
the same exponent, is the same as their product affected wiih 
the exponent. 

By IL, Art. 97, a" -f &« = a« 6-». 

Also, a« 6-« = (a 6- 1)« = (? ) • 

Therefore, a** -f- ft" = [ r 

Similarly, a-" -^ &-»» = f r) Hence, in general, 

II. TAe quotient of any two factors, each affected with 
the same exponent, is the same as their quotient affected with 
the exponent. 

Illustrations: I. 22 X 3^ = (2 X 3)2 = (6)2 = 36 ; 28 X 3^ X 48 
= (2 X 3 X 4)8 = (24)8 = 13824 ; 2-2« X 3-2« X 4-2" = (2 X 3 X 4)-2 « 
= [(24)2]-" ^ (576)-«= ^ ; (f)-2 X (f)-=^X (i)-2z. (| x f X i)-2 



1 



(})- = ^=16. 



16\-8 



II. 242-^62= (5^4)2= (4)2:^16; (_16)-8--(-4)-8=(--^) 



1\4ot /'1\4'» 1 



These examples are said to be simplified, that is, they are expressed 
in their simplest forms. 



EXPONENTS. 261 

Exercise 90. 
Simplify : 

1. (n^f X {a^f X (71)2; (J)2 X (2)2 X (§)2 - (if. 

2. (a;* y-'")8 -f- (oj- V")^ (216 2^2)4^(54 2^-2)4^ 

3. (|a)8 X (|aj-2)8; (^ri)-" X (^■)-; (a;)* x (2:"^)*. 

4. (x)" X {a^Y; (f)-" X (})-" X (2)""; {a-Hf X (a6-8)6 

5. (2 71)10 X (2- im)iO; (a 6-ic-2)3 - (a-ife-ac-*?/^)^. 

6. (4a*^a:«)-"^(2-2a-3*2;-V)"'*; (^-iy*)-^-^(^^rV. 

7. a--- X (3 2>"')-" X (ci)-™; (:r)i- - (^'y-". 

8. («-2 5)-2 X (« ?>-3)-2; (rtS J8 + ^6)- 3 ^ (a6-an3)-8 

9. (i)" X {^T X (J)" ; (a2'' + a" ?»2'')-i x («" - &2«)-i. 

10. («-i)-^ X (xi)-5 X (x-t)-6 X (at) -5 X (&i)-^ 

11. (^^"+*)'' X (i2-*j" X (an)" X {b-nY; aS -^ (2 a)8 

12. « 2x(2a)-2-f.^^y; (2«-2)-2x^^y'x(|a)-2. 

13. (a-i V^x)-' X (.r-2v^6)-8. (,,i)2'' X (2^)2" X (c")2". 

14. (2")-" X (2"-!)-'' X (2-2—1)— X (2-2"+i>-'» X (r)-\ 

15. (2''+i)"' X (2— "+")"• X (2"'-!)'" X (4-"-^)'" -T-(ir))-'". 

16. [(2:-y)-8]-X[(rr + 7/)-]-8; (|)-" x ff)- - (J)- 



252 ELEMENTS OP ALGEBRA. 

99. Positive Fractional Exponents. If m and n are both 
positive integers, 

Kan) = a"*. 

m 

Take the nth root of both members, a'* = \/a^. 

m 

Therefore, a« means the nth root of the mth power of a, or the 
mth. power of the nth root of a. Hence, 

The numerator, in a fractional exponent, denotes a power, 
and the denominator a root. 

The denominator of the exponent corresponds to the index of the 
root. Thus, (81)1 = \/{Siy = (^8iy = (3)3 = 27. 

m 

In a« = /y/a"*, m is the index of the power, and n is the index of 
the root ; also a, m, and n may be any numbers. The expression 
may be raised to the power indicated by the numerator of the expo- 
nent and then extract the root of the result indicated by the denomi- 
nator; or, extract the root first and then raise the result to the power 
indicated by the numerator of the exponent. Thus, 

(-8)1 - -V^FS? = v'64 = 4 ; or, (- 8)f = (-v^^)' = {~ ^Y = 4- 

Notes: 1. a-« is read "a exponent —n;" a" is read "a exponent -; 
a~ « is read "a exponent ." These are abbreviated forms for "a with an 

exponent —n; etc. 

m 

2. It is manifestly incorrect to read a« " the - th power of a." There is 
no such thing as a fractional power. 

3. We must be careful to notice the difference between the signification of a 
fraction used as an exponent, and its common signification. Thus, f used as 
an exponent signifies that a number is resolved into five equal factors, and tlie 
product of four of them taken. 



mXc mc 



100. By Art. 73, a»» = a« x " = anc; 

m 

e 
m 7H-T-C n 

also, a** = a" ^ " = a^ Hence, 



EXPONENTS. 263 

L Multiplying or dividing the terms of a fractional ex- 
ponent hy the same number will not change the value of the 



expression. 


^n^a^-"^. 




But 


a^^^=^a^, 




and 


^a^=^/^a. 




Therefore, 


Jrn - \/;^a. 


Hence, in general, 



II. The mnth root of a number is equal to the mth root 
of the nth root of that number. 

niuBtrations. 

2* = 2* ; 6» = 6i ; 62« = 6«^; ^^64 = '^ ^64 = ^ = 2. 

101. Negative Fractional Exponents. If m and ti are 
both positive integers, 

( --Y 

\a »•/ = a""*. 
By II., Art 97, a-« = ^. 

Take the nth root of both members, 

m 1 

a » = — . Hence, 

Ajiy expression affected with a negative fractional expo- 
nent is equal to the rccipror/U of the expression with a cor- 
responding positive exponent. 

_"* 1 "• 1 
Notes : 1 . From the relation a »• = -^ , a" = — ^ . Hence, the method of 

o* a * 

Art. 30 is true for fractional exponents. 

2. Any factor of the dividend may be removed to the divisor (or from the 
numerator to the denominator of a fraction), or any factor of the divisor to 
the dividend, hy changing the sign of its exi>onent. 



254 ELEMENTS OF ALGEBRA. 

Illustrations. 2^^ = ^ = ^^1 (1)"^ = (^ = | = I' ^ = «"^ 

,,.-1 -3 1 1 1 2* X 3 3 / u^ 

X (i) ' X 4 ' = ^, X I X -3 = -j,~ = - t-i ^x-")~ 

1 11:^, 
-^ -^ - x^ ' x~ x^-^ • 

102. (ah^y=ab. 

Take the nth root of both members, 

J 1 1 1 j^ 

Similarly,* an x &" X c« X . . . . ;?" = (a 6 c . . . . ;?>. Hence, 

I%e product of two or more factor's each affected with the 
sa7ne root index, is the same as their product affected with the 
root index. 

In the same manner we can prove that 

Kotes : * 1. If we suppose that there are m factors oi a,h, c, p, and that 

each factor is equal to a, then it follows that 

By Art. 99, [a^Jn = an. 

Therefore, Va»/ = an. 

2. Similarly, \an) = a. Hence, 

Tfie nth power of the nth root of a number is equal to that number. 

Illustrations. (A)* X (f,)^ X 8* = (f X | 8)* = ^'^ = |; 






EXPONENTS. 255 



»v 



103. (a" X a'T' = (j««+»*. 

m b JL 

Take the ncth root, a« X a« = (0™'+"*) 



m . 6 



By Art. 99, (a"» «+"«>)"'' = a" ''^ «. 

m 6 m 6 

Therefore, a»Xae = a«'^«. • 

iw ft r t ?j.*j.'"i. ? 

Similarly, a* X a« X a» X • • a« = a» « i"*""" «. Hence, 

I. Th£ product of several expressions consisting of the 
same factory affected with any exponent, is the factor with 
an exponent equal to the sum of the exponents of the factors. 

By Arts. 101, 21, c* -f- ac = a* X a « = a»» «. Hence, 

II. The quotient of two expressions coTisistiiig of the same 
factory affected with any exponent, is the factor with an 
exponent equal to that of tlic divideiid mimis tJiat of the 



lUuBtrationa : I. 5* X 5"^ X 5 = 5*"*"^* = 5* = >^125 ; 
X* X a:* X a:" = x"+*. 

II. 2*-r2* = 2^"* = 2i = v^2; (a + 6)' -^ (a+ 6)* = (« + 6)'"* 
= (0 + 6)- A. 

Exercise 91. 

Simplify : 

1. 16-f X 16-i; 25-i X 25i; 3^ x (^; a"! X ^. 

n n 6_»» X 2 m 

2. aixa^Xa^; n-^' X n '; m "Xw ''; 2-iV2. 

3. y"^ X 7/" » ; a' -T- a~'; rt* X J; (a^)* -r- (a^)h 

4. (-2)-i-(-32H; a* -at; (^-J - (..y)-i. 



256 ELEMENTS OF ALGEBRA. 

5. x^ ^ rr2«; a '^-^m ^ ; (a- h)^ - (ah + hi)i. 

6. 32« -^ 3" ; (a - &)" ^(a-b); (x-^/2)h ^ (// 2;^)?. 

\a^b ^J \a-^b^J \ax-^ J \x ^ J 
8. {a -2 xf x{a-2xf x{2x-afx{ci-2 x^. 

10. (.r + yy-'' ^{x + ^)-"; a3^+2y _^ a2^-3^; lA-^rA. 

/ m \ ^ / »« — 1 \ —1— / \ "' + ^" / \ ^ 

11 1*^ Xmnp / 3? \mn/) f CC\ n f CC\m 

13. at -=- ai; 2" x (2»)»-' x 2" + ' X 2»-i x 4-". 

^^2 . i^y {off (x'f . , ; .. , ^ /^V''" 

X04. caa"'=(a")-=a-. 

Take the n qih. root of the first and last members, 

(m r ( \p mp rp tp 

The principles of this chapter are true, whatever the values of 

a,h,c,....m, n, p, and q ; that is, a,b,c, m, n, p, and q can be 

positive or negative, integral or fractional. 



EXPONENTS. 257 

niustrations. (2' X 3* X 4"^)' = s'""* X S*""' X 4-*'** = 2« 

X 2* X 3* X 4-1 = ^2 X V3 ; • L(a"^)"«]fl^ -r- I [(aW^JJlJ 



N m 



a"« »• = a-* — a = a. 

105. Negative and Fractional Root Indices. 

_ j>i_ _w \ 1 

an V« 

m _m 

_ e _£ £ 1 1 

Similarly, y^a™ = a»« = a « = — = "^TZ* Hence, 

^ negative root index, either integral or fractional^ indi- 
cates iJie reciprocal of Uie expression with a corresponding 
positive index. 

Note. Since it is impossible to extract a fractional or negative root, or raise 
an expression to a fractional or negative power, in order to perform the opera- 
tion indicated by such indices some preliminary transformations must be made. 

lUuBtrationB. ~i/^ = -: — = -| = -r = —^ ; i/4a« = (4 ay 

_ I _ ± ^_ _ ± _ J- = 1 

a* 

Exercise 92. 
Simplify : 

1. 1^27; V^; |/32 m-iO; Vsla-^; V^. 

2. 1^8; [(63)2(a*)8(6-8)(a-6J-i)2]6; ^8a*6— ic"-2 

17 



258 ELEMENTS OF ALGEBRA. 

2 

V ^ / \y'J Kni^n^J ' vo«'"^'*^"^' V25' 

1 1^ / J_ \a2 - 62 

Queries. What does a negative exponent indicate 1 A fractional 
exponent ? A negative fractional root index ? Any expression with 
for its exponent = ? Why ? What is the product of as and a^ 1 
Prove it. 

Miscellaneous Exercise 93. 

Express with fractional exponents and negative power 
indices : 

1. ~\^^; ~^; 4"^; 'V^; (^a)^; ^^5^2 

m 

Express with radical signs and negative integral root 
indices : 

m O 

3. a-t; a^hic~^; 4:ah~^; 7a~^x~''; — - . 

X 4 



EXPONENTS. 259 

Express with radical signs and fractional root indices : 



n n 



9 1 m m X T7"« 

4. at; (4a2)!; alz^U; a'"5"; a" ft"; 



xy. 



Express with fmctional exponents and fractional power 
indices : 

5. ^Jbi; y/2'6-; 3v'(8a-8)-|; ^a'~^; Va; ^^5. 

Express in the form of integral expressions : 

Sa^b 5 m* a x~^ x~^ a^ 



c-2 ' ahc' n-r 4^1' 4^^' -^-|' aib-l 

if 

Express with literal factors transposed from the numera- 
tors to the denominators: 

Simplify and express with positive exponents : 

8. 4^; v^^^a. yj(h^; "-^/^; «-»; [Va^ H- V^]"". 

_ 2 ai X 3 a-i a 2^ x a'^ x v"^ «/ ^ s.-y 

9. -== ; —=r. — ^^ ; Vm-3 -^ V7/il 

10. 2;-ix2a;-i; (^V*; a^ x «i X a-J. 

11. -^4^; (j^y^; V^^h X ^J'^F-s 

12. aUiaxa-i6-U-i; (f^)"'; y/^- 



260 ELEMENTS OP ALGEBRA. 

13. aH^c^ Xa-H-^c-i; \l^\ i/(^~H*)^. 

14. y" X y X "; (x + y)^ X n^ X n~r^ X Vti. 

15. Y/(m^'^V'; y/«»6"i i(|)"'; -tF^. 

16. (77i-i\^a)-3 X 'v/(a-2 V^"; y^^' + — 

17. VaF^W^'-^{ah~~^y; l^^ll^. 



2"' 2»2 



a; '■m "^ n 



^^- m2 ^ m-w Ml , J' (m2-7^2)2 
2"(2"-i)" 1 2" + ^ 4""^^ 



2»+i X 2"-" 4^"' (22»)"-i • (2"-i)" + > 

(9"x32x5kV27» 

V / / ; (2" X S"*)"" Is" X 6*"")"' 



23. pir^^ 



EXPONENTS. 261 

Multiply : 
24 a*fe~i - a^h"^* + 1 by a^h~^' + 1. 

25. a^ + a2»&*' + h^' by a" — a^"^)*"* + ^'. 

26. a^h '—a* 5 •+« *7>i— a »6« by a«6 i + a "ft*. 
Divide : 

27. a^ + rt^ M"' + &• by a» + a2^ M"' + ft*"., 

28. a,-*"*"-*) — y2m(m-l) ^y ^{n-l)^^(m-« 

29. a;**"* - if^-* by ar"*"'' + i/^'"-^ 

30. a;^"'-*" - ^m'-am by ^"^-^"^ ± f*-'*, 

31. a^ —3^+4:a*'*x*'—4:a^x^' by a2* + 2a^2:<'' — .r^*. 

32. a3+a~i* by a5 + rt-^; riT/i + mx^ by n^yi + w^a;^. 
Separate into two factors : 

33. a-^-b; a'^ - ft-f ; aV - 6-2«. 

Expand : 

34. (a-U-6-ia:)*; (a:-2a;-i)8; [(a"^ -«?)']". 

Resolve into prime factors, and find the products of: 

35. N?'!^, 4^2, \/96, \^ 

36. ^12, v^72, \^, ^, ^^^576, V2l 



262 ELEMENTS OF ALGEBRA. 

Find the cube roots of : 

38. 8 a-2 - 12 a- V- + 6 a-f - a"! 

39. a;3__9^+27a:-i-27ar-3. 
Find the 6th roots of : 

40. a;« + ^ - 6 (a;* + ^) + 15 (^x^ + i) - 20. 

41. 729 - 2916 a2« + 4860 «*" - 4320 a^" + 2160 a^" 

-576^10" + 64a^2«, 

42. a;- 12 - 6 :r- 10 + 1 5 a;- 8 - 20 ^^- 6 + 1 5 ic- 4 - 6 aj- 2 + 1. 
Simplify and express with positive exponent : 



!i+i, ifan i/4 X 4"-i 

H:i, 

y/4«-i >^ 4„ + i 






4fi ^"^"^ [(8a-6?>)2"]5" Q 

9(a;0 + 2/0 + ^)-2m3' [(4a-3 6)5"]2''' ^ + T • 

.,, (20a3H8a;2?/2-12y^)" (m^ + ^^)^ (^^ - n^)^ 
[4 (2:2 + 2/2).f ' ' m6--?i6 



RADICAL EXPRESSIONS. 263 

CHAPTEE XIX. 
RADICAL EXPRESSIONS. 

106. A Surd is an indicated root that cannot be exactly- 
obtained ; as, V5 ; 'V^f ; \^a^. 

The Order of a surd is indicated by the root index. 

Surds are said to be of the second^ third y fourth, etc., or nth order, 
according as the second, third, fourth, etc., or nth roots are retjuired. 
Thus, 'v/a, ^a, \/b, etc., yx, are quadratic, cubic, biquadratic, 
etc. 

Surds are of the same order when they have the same root index ; 
as, ^b, ^a\ and ^¥. 

A surd is in its simplest form when the expression un- 
der the radical sign is integral, and in the lowest degree 
possible ; as, ^32 a* = \/2^ a^ x 4 a = 2 a v^4 a. 

Similar or Like Surds are those which, when reduced to 
their simplest forms, have the mme surd factor ; as, 3 \/3 
and a/3 ; 2 a vh and c ^/h. Otherwise the surds are 
dissimilar. 

Hotes : 1 . When a surd is expressed by means of the radical sign, it is 
called a Badical ExpressioxL 

2. An Irrational Expression is one which involves a surd ; as, V3 ; 
a -\-h \c^. 

3. An indicated root may have the form of a surd, without really being a 
8uixl. Tims, Vi and Va» have the f(rrm of surds. 

4. Rational factors or expressions are those which are not surds ; as, 2; 
a*x — bf^y. 

5. Since a" — a^P, surds of the form Va^ and fo^ are equivalent sards 
of different orders. 



264 ELEMENTS OF ALGEBRA. 

6. A Mixed Surd is the product of a rational factor and a surd factor ; as, 
a V6 ; 3 Vb. 

7. An Entire Surd is one in which there is no rational factor outside of 
the radical sign; as, V2; \'a^; Vx. 

8. A binomial surd has two terms, and involves one or two surds; as, 

a -\-b Vx] a Vx — b yy- A compound surd or polynomial has two or more 

2 3 - 4 - 

terms, and involves one or more surds ; as, y2 + 3 4/4 — 5 V3 ; 

a-\-h- c + 2Va. 

9. Quadratic surds are of most frequent occurrence. 



107. The methods for operating with surds follow from an appli- 
cation of the principles of Chapter XVIII. Thus, 

f = V^f . 2 a2 &3 == ^(2^2y3y3 ^ ^^^;^9; j^ general, 

n 

a = a^ = a^— ^a^. Hence, 

I. To Reduce a Rational Factor to the Form of a Surd of 
any Order. Raise it to the power indicated by the root index, and 
place it under the radical sign. 

2V'3 = V2' X 3 = Vl^. f ^9 r= ^(1)3 X 9 = ^|. In general, 

n \ 1 

a ^x = an xn — (a'*;r)» = y'a^. Hence, 

II. To Change a Mixed Surd to the Form of an Entire Surd. 

Reduce the rational factor to the form of the surd, multiply by the 
surd factor, and place the product under the radical sign. 

V72 = V62 X 2 = 6 V2. ^1029 a* = (7^a^ X Says = 7 a ^3a. 

9 3/7 _ 9 i3/iZi _ ?Jl1 ,3/T ^.VJI _ «,V 1 X^'« 

^Vu-^y2x4~ 2 - V4. 2V2a3 - 2V2a3 X 23a 

a / Sa ^j — 

"^ 2 V 2^ = i V 8 a- In general , 

Hence, 



RADICAL EXPRESSIONS. 265 

III. To Reduce a Snrd to its Simplest Form, ii tiie surd is 
integral, remove from under the radical sij^n all factors of which the 
indicated root can be exactly obtained. 

If the surd is fractional, multiply its numerator and denominator 
by such expression that the indicated root of the denominator can be 
exactly obtaiued. 



\/2^ a X v^o^ = \^'2^a x a^ = a \/2. In general, 

_ — — A J * * * 

^a X ^/b X ^c X . . . . ^ = a'* X b" X C^ X . . . . p^ = (abc . . . . py 
= \^a be p. Hence, 

IV. To Find the Product of Two or More Surds of the Same 
Order. Take the product of the expressions under the radical signs* 
and retain the root index. 

In general, 

^/^-^\/r=(^y=\/'f- Hence, 

V. To Find the Quotient of Two Surds of the Same Order. 
Take the quotient of the expressions under the radical signs and 
retain thn root index. 

f/i5^64 = ^64 = 2. (/ V25^ = 1^(2»)'' = 2^ = 4. In general, 

^'^ = (""j* = a^ = "^a. Hence, 

VI. To Find the vith. Root of the lith Root of an Expres- 
sion. Take the mnth root of the expression. 

Note. It is sometimes easier to j>erfonn operations with .simls if the arith- 
metical numbers contained in the surds be expressed in their prime factorSf and 
fractional exponents be used instead of radical signs, 



266 ELEMENTS OF ALGEBRA. 

Exercise 94. 

Express in the form of surds of the 3d and nth orders, 
respectively : 

1. 1; |; 22; 4"; 2 a"; Sahc; S x; a^; of; af'y\ 
Express as entire surds : 

2. JV2; 1^3; 5 V32 ; f V^; leVflf; abVbi. 

3. a 4/d^ b 6-8 ; 3 a^ ^ofc^ ^ i '^^ ; 2x</J^; | ^. 

5. 5-;.^^25^i; (,»-l)v/^; '^^±^\I^^EI . 

' ^ m — 1 771 — 71^ m + n 

^' ;r^V-^r^' ^V^?^' ^"V^r-; ^^Vs^' 

Express in their simplest forms : 

8. -^288; 3V150; •^^^^IIS?; fV90|; 2 a^W^. 

9. V3i; ^Jl|; ^J^; ^'1029; Vf; V^ ; ^|- 
10. <1\ ;J|; ^=T08^a, ^3^i;;i5ro, ^Z'^- 

11. ^^?n=^^\ ^'V^; !^v/— • 



RADICAL EXPRESSIONS. 267 



. -^', ^7290a3-j6m^2. ^^J^; ^a^^+Y"- 



12. ,/ 

^586 



13. V(a; + y) (a^ - ^) ; Va«2 - 8 aa; 4- 16 a. 

14. ^^yJtlEI^LlIl, ^ 1715 ^^-V^ 

Simplify : 

15. Vl2 X Vl8 X V24; V54 -- Vl) V^Vlf. 

16. ^Fex ^^54x^/128; [v0[28^ -h ^5^6^] ^ >^9^. 



17. ^v^^:^.- X 



V50a8 66^ V32a63 



18. V2^ a8 ^6 X vOUe a2 m2 a:^ X v^56 a^ m^ x\ 

19. (^53a«fe9 -4- -^25 a* 6^) x ^125 aH X <^W^. 

20. (V6M -^ V63~?) X v^54^ -T- v/feT: \/'^^^^^. 

21. (^iiT?^ X ^a-ift-ic) -^'(v^a:-^oyo x ^^lO^X 

22. (^|^^)^V20736; 'i^ivF^ - aX^. 



23. Vf a8 X Vf a-2 X V.f ai X V2.5 a"*. 

24. \1\J </W^^^\ (16 aH2)i X (ai h^f -r- ^2 J h. 

108. ^5/2 = •^^21^ =^8. 3^=3*^]|^2^M? = 3]!J^l6. 

In general, 

p pxw 

y^aP = «" (n > /)) = a* »< "• = "^aP"*. Hence, 



268 ELEMENTS OF ALGEBRA. 

I. To Reduce a Surd, in its Simplest Form, to an Equivalent 
Surd of a Different Order. Divide the required root index by the 
root index of the surd, and multiply the power and root index by the 
quotient. 






TheL.C.M. oftheroot 
indices (3, 9, 6) is 18. 
In general, 



pm 
Pin 

^6w = fe"*" (m > pi) =: "^6p.». Hence, 



II. To Reduce Surds, in their Simplest Forms, to Equiva- 
lent Surds of the Same Lowest Order. Divide the L. C. M. of 

the indices by each index in succession. Multiply the power and 
root index of the first surd by the first quotient, of the second surd 
by the second quotient, and so on. 



Exercise 96. 

Express as surds of the 12th order: 

1. A^2; ^3; f^; 3^2; ^a^; ^1; i^^S. 
^ 2. a/sS; 1^32; ^a^; v'^^X V^^^i"^. 

Express as surds of the 7ith order, with positive expo- 
nents : 

3. ^x^; V^; ah; ^'^j}; -L; v/«~"; ^. 



RADICAL EXPRESSIONS. 269 

Reduce the following to equivalent surds of the same 
lowest order: 

4 V5, ^11, 4^; a/2, \^5, \/3; \^8, V3, ^6. 
5. ^2, ^8, ^i; v^7, ^5, ^6; Va, ^a^ 
G. '^^, Va; ^^«, ^a6, ^a^ ^1^?, '^^^. 

8. Vaic^, ^/a^Q^\ ^fm, "^n, v^, ^mnx. 

9. v"^, \^6^ ^;?; 41^5^, 2^VlW^, 10 a a/37. 

109. fV6=A/(IF^^=A/i =a/W. 

i a/5 = a/(|)'' X 5 = V¥ = a/H .-. IV'5>Ia/6- 

In general, 

_ J 

a J^x — (a" x)" = y^a* as, 

5 ^'y = (6- y)* = .y^Fy. Hence, 

I. To Compare Surds of the Same Order. Reduce them to 
entire surds, and couipaie the resulting surd factors. 

\ ^52"= ^{\y X 2« X 13 = y ^' =^^/42:25, 

I ^8 = ^{\y X 2 = ^ (f)« X 22 = ^45.5625, 

3 Vl = a/3* X f = '^3« X (fli» = >^46.656. 

Therefore, the order of magnitude is 3 ^\, \ ^, \ >^52. 
In general, 

— ??J? JL 
6 y^y = 6»»» ir ■ = "v^b^-y*. Hence, 



270 ELEMENTS OF ALGEBRA. 

II. To Compare Surds of Different Orders. Reduce them to 
entire surds of the same order, and compare the resulting surd factors. 

Exercise 96. 

Which is the greater ? 

1. 3 V6 or 2 Vl4 ; 6 Vll or 5 VlSf ; 4 VG or 6 Vi 

2. 10V5or4V3l; iVTorfVlO; ^^2 or ^3. 

3. V|or^l|; ^4 or ^5; Vf or ^T|. 

4. ^11 or '^f; 1.6 or J ylO; \^6^ or V^. 
Arrange in order of magnitude : 

5. V3, </4, </7 ; 8 V2, 5 a/5, 4 V7|. 

6. 2'^2l, 3^49, 4V7; 3^4, 4 ^I|, 2^131. 

7. 3 '^2, 3V2, |A^4; 2^21, 3^8, 2V8. 
Show that the following are similar surds : 

8. ViO, V90, Vf ; J V'20, i V45, 5 Vf. 

9. 7 V|, 'V/ff , 3 VS ; -^162, 3 ^32, </2E92. 

10. V27, Vr92, Vl47, Vl; a^W^^ h</W^^ f V— . 
11 QiV*^ ^V5T2 ^K^l^- i/«^ .V«*^'^ ^.la^(?m^ 






RADICAL EXPRESSIONS. 271 

110. Addition and Subtraction of Surds. 



iVH"^=-iV^|x2 = -fV2. 



Adding, 



3»a«Xa^arx(26)g 3 a o/-—— 
26X726? =26^^^ ^ ^' 



3/27a«x s/ 

-°V 26- -y-2hxm = ~ 26 ^^^«'*'^' 

1 8/46^ 13/4 62a: X a* 1 3/ — sttt- 

6V^ = ^V^=^3r^^ =^lJ/4a262x. 



Adding, 

= ^^-y- ^4a^b^x. Hence, 

Reduce each surd to its simplest form. Prefix the sum or differ- 
ence of the rational factors to the common surd factor of the similar 
surds. Connect dissimilar surds by their signs. 

Exercise 97. 

Simplify : 

1. 3 Vis - 2 \/20 + 3 V5 ; 3 \/| + 2 V^. 

2. 2V| + 3Vi; 2^l62-J^^^; 3 V^ + ^?^. 



272 ELEMENTS OF ALGEBRA. 

3. A/3 + Vli-2V5i; 5v^^=^54-2v/::r6 + -i'^685. 

5. S^J^+^Ml-W^; 3\/l62-7^!^32 + ^1250. 

• 6. ^?+ 1--^^- 3 V^27^2. ^40-3 v''320 + 4^^135. 



7. V50ab^c^-VS2aH^- (4:hc^-3ac)V2al)c. 



8. a; Vwi^^ 71^"^ x^ — m vm^ n^^ x^^ + n -^m'^ n^ aP. 



9. V3 a Z^'^ + 6 a& + 3 a + a/3 a &2 _ (3 ^ ^ + 3 ^^ 

10. y/^+Y/^_2« + «^-^V^^^. 

^ a — ^ a -{- a^ — h^ 

11- |^A + 0-SVf-iVV96+1.5^|-ii^T750 + 8V|. 

111. Multiplication of Surds. 



3 -v/a X 7 V^ = 21 y4 X 3 = 42 y'3. 

/^2 X ^3 - lJ/2^"xT3 = :^432. 

f V2 X 1^3 X lA^ X -^i = f X I X f ^£6 X 3^ X {\f X (i)« 

= <{^28= -^2. In general, 

a ^x X h y^y = a x"" X h y^ = ah {x ?/)« = a 6 -y^ari/. 



a yar X h 'J^y = ax"" X b y'^ = a &(»;'"?/")"'« =: a & y'a;"' i/**. Hence, 



RADICAL EXPRESSIONS. 273 

I. To Find the Product of two or more Monomials. Reduce 
the surds to the same order (if necessary). Prefix the product of the 
rational factoi-s to the product of the surd factors. 



Multiplicand, 
Multiplier, 


3V3, 
2^6, 


3V2- 
3V3H 
9^6- 

9V«H 


-2>^5 
h2^ 


3 >y/2-2v^5 multiplied by 
3/y/2-2'v^5 multiplied by 

Sum of partial products. 
Hence, 


-6^5<^X3» 

6 ^2«X 62-4/^3(7 
h 6 v^288 - 6 -^'675-4 v^30. 



II. To Find the Product of two Polynomials. Proceed as in 
Art. 24. 

(^^/2-\-2^3){^^2-2^/3) = (3 X 2*)^-(2 X 3*)^^ = 32 X 2-2^ X 3 = 6. 
(a^x-^b\/~y) (a^x-bx/y) = (a x^^ - (hy^f = a^x-h^y. Hence, 

III. The product of the sum and difference of two binomial 
quadratic surds is a rational expression. 



Exercise 98. 
Simplify : 

1. 2'v/r^ X 3 V3; SVf X J\/T62; J VlO x J^ Vl2j. 
2\/l4 X V2i; 3^1 X gVJ. 
(5 V3-5) X 2 V3; \!^64x2V2- 

4. (V2 + V3 + 2 \/5) X V2 ; 4 ^75 X 2 V^. 

5. J V4 X v^iO; i VJ X § V^l; V5 X ^2. 



2. J ^4 X 3 v^2 

3. 3 >^3 X 3 V2 



6. 3 \/| X ^1 ; J \/§ X 9 -^1 X \^. 

18 



274 ELEMENTS OF ALGEBRA. 

7. 2^3 X'v/2 X J'^i; V^%X</'i; ^T68 x '^147. 

8. '^^X^9X^9*; (3V2-3'v/6-V8+3V'20)x3V2. 

9. a/5 X V^IO ; (Vn - Vm) xVn; 4 \/^ x 3 VS. 

10. Vmn X \^S m^x X V2 nx. 

11. "^/m^no^ X "^m^ n x ; 2 Va X '^^ X 3 ^a x '^^. 

12. ^^2^ X ^3^ X V -i-; ^^(4 7/^ a^'^)" X ^(:2m^x)\ 

13. V-xtV^r^; (V2-3V3)(2V3 + 3V2). 

71 ▼ 71 3 ▼ 2 ft* 

14 (3V5-4^2)(2^5 + 3V2); (^2 + ^3)^. 

15. (5 V3 - 6 a/2 + a/5) (lO a/3 + 12 a/2 - 2 a/5). 

16. (V2 + </l+ '^D «/2 - V3); '^24 X 6 ^3. 

17. (V2 + V3) (V2 - VS); (^3 + ^4) (^3 - m 

18. (a/5-a/3)(V5 + a/3); (a/5 + 2 a/3) (a/5 - 2 a/3). 

19. y^l2 + a/19 X y^l2- a/19; v'TG X a/S. 

20. y'9 + A/n X \^9 - A/rZ ; a/3 X ^2 X ^|. 

21. (^a3 + ^^) (^-2 _ ^-3) . y| ^ ^1 



RADICAL EXPRESSIONS. 275 



22. V^10+ V68 X y^lO- V68; -v^^^xy^^. 

23. {dVx+3 Va-^x) (5 Vi - 3 Va-2^x), 

24. ^S X ^M; VW X ^i; {mfi<^Mf' 

112. To Rationalize Surd Denominators of Fractions. 

2 2 X V3 2 V3 2 
;^ = 7^^ -7- = -^ = T^ X 1.732 + = .23 + . 

2 2X -v/3^« 2 ^3« 2 ,/_ 



— =r= = - — !^, = — - — (n>m) = — T . Hence, 

I. If the Fraction be of the Form — — = . Multiply both 
terms by y"^^^^- ^ ^^ 

3+ V5 _ (3 + ys) X (3 + ys) _ 3« + 2 (3) ( /y/s) + (^5) * 
3 - V5 ~ (3 - V5) X (3 + Vs) ~ 32 - ( VS)^ 

14 + «V5 7 + 3X2.236+ ^ ^^ 
= 9-5 = 2 = ^•®^^"^- 

4V3 + 3a/5 (4 a/3 + 3^5) X (2V7-3 V2) 

2 V7 + 3 >v/2 ~ (2 V7 + 3 V2) X (2 a/7 - 3 v^ 

_ 8 \/2l + 6 a/35 - 12^/6 - 9 a/To 

g ox(A/^TA/g) _ aiVbTVc) _ a{\/bT\^c) 

V^±A/^"(A/^iA/^)x(A/ftTA/^)" W~-W~ ^"^ 

g a X Tfe T A/g) _ ajh^^c) _ a(hT Vd „ 

b±^~ (b±\^c)x(bTV~c)~ (py-ic^^ " ^* ~ *^ ^"^' 



276 ELEMENTS OF ALGEBRA. 

II. If the Denominator is a Binomial Involving only 
Quadratic Surds. Multiply both terms of the fraction by the 
terras of the denominator with a different sign between them. 

Note. It is often useful to change a fraction which has a surd in its de- 
nominator to an equivalent one with a surd in its numerator. Thus, 

8 SXVI 8 I'S^ J X 2.236+= 1,3416+. 



V5 V5 X V5 5 

Exercise 99. 

Eationalize the denominators of: 

2 3 2 - ^2 3 V5 



1. 



V2 + V3' 2 V5 - V6 1 + V2 V3 + V2 

8-5 V2 , 2 V"5 - V2 1 6 

3- 2 a/2' V5 + 3V'2' 3-2 Vg' 'V^64 ' 



Vx — Vy , Sx — Vx y _ Va + a; + V<?' — x 
o. 



"s/x + Vy Va: y — 2>y Vet -\- x — Va 



X 



. X — Vx^ — 1 a 1 2 a 

4. 



+ Vx^-l' \/a+Vb V5-V'2' 3a/2^-^ 

Given V2 = 1.414, V3 = 1.732, V5 = 2.236 ; find 
the approximate values of: 

5. ^_; V50; 8K288 '' ' ' 



6. 



V2 ^ V5 2V675 V500 

1 + V2 1-V5 3 1.1 



2 + ^/2' 3+V5' 21/2-3^/3 ^5-^2 2 + ^3 



RADICAL EXPUESSIONS. 277 

113. Division of Snrds. 

2 V54 -^ 3 ^6 = I VV = f X 3 =2. 

Ingeneral, a^i^6^y = ^^)" =^Vy* 

I. If the Divisor is a Monomial Reduce the surds to the 
same order (if necessary). Prefix the quotient of the rational factors 
to the quotient of the surd factors. 

,- . ,- ,-x 3\/3 3 V3X (3^3-2^2) 

3 V3 -r (3\/3 + 2 V2) = :^ p = 7 7= V\ / y A 

^ ^ ^ 3'v/34-2V2 (3V3+2-v/2)x(3V3-2V2) 

^27-6V6 Hence, in general, 

II. If the Divisor is a Binomial Involving only Quadratic 
Snrds. Express the quotient in the form of a fraction, and ration- 
alize its denominator. 

^a -^ \/b - \/c) a + 2 \/ab + b-c {\^a+ \/b-\-^c. 

Divisor multiplied by '\/a, a + \/a b — \/a c 

First remainder, \/a 6 + 6 + V** c ~ c 

Divisor multiplied by y'ft, \/a b + b — ^/bc 

Second remainder, y'a c + ^b c — c 

Divisor multiplied by \/cj ^ac + \/6 c — c 
Hence, in genenil, 

III. To Divide a Polynomial by a Polynomial. Proceed as 
in Art. 33. 



278 



ELEMENTS OF ALGEBRA. 



Exercise 100. 

Simplify : 

1. 21V384^8V98; 5 \/27 ^ 3 V24; \/l2^V^24. 

2. - 13 VT25 -^ 5 V65 ; 6 Vl4 H- 2 ^21. 

2V98 ' 7 a/22' 5V112 ' V394 ' ^2 * Vs' 
4 IJ ^2| - I Vll; -^12 -- ^2 ; V6 -- A^4. 
5. 20 ^^200 -4- 4 a/2 ; ^18 ^ a/6 ; 4 ^32 ^ '^IG. 

3 a/108 5 a/14 15 a/84 

7. (15 a/105 - 36 v'lOO + 30 A^81) -^ 3 Vl5, 

8. '^OOei^ViO; a^'a^c-^^^; Va -^ \^. 

m — 71 ^ m — n ^ {m ~ nf '3 »2 

10. {acx^Vy—hcy ^/x) -^ c a/^ ; '\/a~x -^ ^'o^. 

11. ^4 m 7^2 -H V2w3^; v^2W^ X ^^^?^3^ aA;?^5 

12. A^4^i2^XA^'9^^i2^*^v'25^^2^; <^d~^-r-\/^' 

13. -y— ^x-V/-2--^— -V/-|— • (aj-l)-f-(A/aj-l). 



RADICAL EXPRESSIONS. 



279 



14. V10.4976 -^ 2 Vo ; (2 a: - Vo; y) ^ (2 Va: y - y). 

15. (:^ a/3 + 2 V2) ^ (a/3 + V2) ; 4 ^a^ -f- 3 Vo^. 
,^ 2A/T5 + 8 . 8V3+ 6a/5 8-4\/5 . 3a/5-7 

Id. =:r- -7- — —', 7^7- -T- pr- • 

5- a/15 5a/3-3a/5 1 + a/5 5 + a/7 



17. (^x« + ^^v + w") - (V'^'^" - va;V + vy). 



114. Involution and Evolution of Surds. 

l^v/ir=[Mi)T=i-.x®'=4v/I=^vi.- 

y486av/4a« = [3« X 20(220*)*]' = [3^ X 2^a^f = 3 X 2*0* = 3^/2^ 

m mp 

In general, (a^i -v^t^)^ = (a«i 6"*)' = a"!^ 6^ = a^h^ >v/6*^. 
Vo'»i v^6™ = (a"*! 6" j^ = a »• 6" ^ Hence, 

Express the surd factors with fractional exponents, and proceed as 
in Art. 104. 

Example 1. 
V A«/;5 2a) ~ Ul 2aJ 

-©■-3Gr(i).3@C4)'-e)' 
-^ -'S)©"K)(^)- 



^? 



2»a« 






3a* 

2cH 
3q« 



3 



4a*c* 
3 



i«ra 



8a» 
8a«" 



280 



ELEMENTS OF ALGEBRA. 



'i 

CO 

+ 



I 

+ 

+ 

-0 

Li 





i<5 




^ 




CO 
1 




1 

1 y 


1 


'> 


1 


^ 




+ 


+ 


-"« 


Ol 


CM 


II 









Th 




+ 


Hw 


-* 


« 


c 


Tt^ 


^ 



I 





HM 




-O 




CO 

1 


^ 


% 


GO 


GO 


+ 


+ 




Hn 


c 


c 


-^ 


Tf 














'-Ca 








^5 


















a 


r^ 








'd 








S 








S 






c 


^ 



1 






fn'' 






fi" 


.2 




-U 




jT 






p-T 
a; 


J 








^h" 








'IS 




ir^ 


-M 


c 




s 
s 


^ 


(V 
S 


id 


1 




S 




1 

i 




s 


s 


-tJ 


C 



8 


-^ 


^ 


-TIJ 


-n 


-s 


02 


CK 


03 


^ 


rt 


a 


G 



rt 


^ 


^ 


ti 


^ 


£ 



















•>-( 






o; 




S 


£ 


s 


pR 


pR 


'/2 


m 


02 


02 



RADICAL EXPRESSIONS. 281 

Exercise 101. 
Find the values of the following : 
1. mf; ^VE; i^Vlf; ^^2; (^32)'. 



2. 'fe^; ^m-, i'^I^jf; V-^Gi. 

3. {</uf; vQ^; mf; ^vff; mt 

4 ^JV^; (^27)^ --(^A; (2^3¥f ; 'sp^- 



5. (2 ^^6)'; y'V©"' '^"•^^■^"^ (2a^2Fo/. 

6. ~^/•i-•a-'■, {Z'^Wc^f; ^iU^"; "v''27»=^. 

7. 'v'»-»v'^»; [\/(a-c/]"; "-^1; [jv^I^^-^js, 

11. v^9-a:— ; [{x ^ y) V^y^ - VnS^2i^^, 

Find the values of the following, and express the results 
in terms of positive exponents, by inspection : 

12. (^^^T?f ; Wl + V\f; {V2 + Vsf. 



282 



ELEMENTS OF ALGEBRA. 



13. (^3^l)MV^+^i)- ["^-fl- 

15. M^.o]^ [(-^«--f +(fr^)-T- 



16. 



[jn^ J'Zm 2-^m T ["^ /m-^ 4 -^^3a "[ 
12 mV n "^ 4/n J ' L V ^4 ^^^^^3 J 



Find the square roots of : 

17. ^ + 1 + 9 ^^ + 1^-1 _ 1^-1 ^^ _ 6 ^^. 

18. 1 - 2"+^ + 4"; 9" — 2"+^ X 3" + 4". 



19. 



r 






V2. 



ic 4?/ -^a; '2/ 

m 

20. v"^ - 4''^2;5- + 4 :i:"^ + 2 a/^^ "' - 4 '-v^^^ '" + a/^' 



Miscellaneous Exercise 102. 

Find the values of the following : 

1. 'V^; v^; V3xV27; 1""; "'^4; a/X^V|. 



2.' ^^32^; V^e^m^^ V^m ^ V^ 
Reduce the following to their simplest forms : 



18|' -^'2'- 



3. t J^,; V^; ^3888; ^!^±^sj-^ 



») 



RADICAL EXPRESSIONS. 283 

Reduce to equivalent surds of the same lowest order : 

5. V2, '^, •^5; -v^y, a!^8, 4/5; 3^75, 3v^54. 

6. 2A/I8, ?,\/U, 4A!^r62, 5-^128; V^\ </V, i^^. 
Change to similar surds : 

7. •^27, -^144; -C^Si, 3^; W2, ■^243. 

8. ^02, VI; VTo, 4'Wb; I'^lh, V^. 

9. 2\/^. ^7-26^, ly^^; K.1G; 4^275. 

10. n, VJl, ^i^, 3 ^^; V20, 3 '^. 4 Vl25. 

11. ^32. ^128; VM^. y/^'. SfW^'- 

12. ^J^. ^192, ^S". ^^. ^A; ^2, ^2?. 
Arrange in order of magnitude : 

13. 3 a/2^, 4 </^, 3 -^3 ; 5 ^8, 3 ^9, 3 VlO. 
14 fnfV3~, iV5, 2^; Vi ^H- 

15. a' -e^^sTjy, -^ ^(25^^-, (64)" sj"^, . 

Find the values of: 

16. V243 4- \/27 4- v'48; 2 V^189 + S'l^'STS -7^, 



284 ELEMENTS OF ALGEBRA. 

17. 4 V5 X 'V^lT; 3 'v'^gOO -^ \/5; \J2 V 2Vl -^ ^V^^' 

18. 5V'2 + 3V8-2V32; 3 -^^ST - 4 ^1^192 + ^648. 

19. 'V^m X #432 ; I V5 X |- #2 x #80 x #5. 

20. ^64+5-^32-^^108; i(^27 + | -V^i92 + •^81). 

21. 2'v^JJ^ + a/60 - \/225 - Vf ; J V|^ { a/2 + 3 V| )• 

22. (a^|^2^^)--V^I6; (-^9-2^21 + 4^1)2-^9. 

23. (6 -^1 + ^18) -f- -^72 ; 1| -^/20 - 3 -v^5 - -v/f 

24. J,a ^60-^(2 ^240 + 7^31); y/j^-^^^)*. 

25. (-^ro -2-^4 + 4 -^54) (o -^64 + sV^- 2 ^32). 
Eationalize tffe denominators of: 

^g V2Q - a/8 . (3+ V3)(3 4- #5) (#5 - 2), 
* V5 + V2 ' (5 - V5) (1 + VS) 

^ m -\- (171 — Vvi^ + a? 71^ x"' 2 



m — a ?l + V??i2 ^- ^2 7^2 „ ' ^5 _|_ /y/3 _ ^2 

Simplify the following : 

28. ^_ ; V(f|y" X V(||f ; #(8'a3&)2 x #(2 a &3)2. 

V2 

^^ 7 + 3 \/5 _^_ 7-3a/5 ^ /^9?»Ti x V3l<T" 
^" 7-3^5 7 + 3A/5' V 3V3^ 



30 



. {a -r #«)" + ' y^(^ j/^"")"'; #^2^^""* X \x^yl 



RADICAL EXPRESSIONS. 285 



31. 4^aX'i^:r^X ^a* X "/a X ^aV x ^^. 



33. 4.x^<77xf x-^^^„-^(^y-^^/*^ 

34. V(a>) ^ v/(-^.,p; 1^,; |(|)t. 

35. .^- (g) ^ Vl^. ; I ^1 + 1 ^^' - 2 (S)«. 

37 V^ + «^ V^ ^^ ^ 4 ^^ 

38. ^-Hl^^^i -H f 1 + -^V; i^a^ + ^a)3. 

^5 X ^^3 
40. Express r^ with a single radical sign. 



Queries. What sign is given to the Titli ])ower ? To the nth 
root? Why? How change the order of a suixl ? In T., Art. 112, 
why take m less than n ? How rationalize a sunl denominator ? What 
powers of n^ative nuniWrs are positive ? What n^ative ? 



286 



ELEMENTS OF ALGEBRA, 



Imaginary Expressions. 

115. An Imaginary Expression is an indicated even root 
of a negative expression ; as, V— a ; a -\-h V— 1. V— 1 
is an imaginary square root ; a V— 1 is an imaginary 
fourth root; etc. 

-V^T^^ :^ ^a2 X (- 1) = V«^ X V^ = « a/^^- 
/^-TJ - .^6 X (- 1) - ^b X V-^- Hence, 
Every imaginary square root can he expressed as the product of a 
rational or surd factor multiplied by \/— 1. 

The successive powers of ^y/— 1 are found as follows -. 

)ip=(-l)i = + V=T; 
)*/=(-!) =-l; 

)*r = (-i)' = (-i)(-i)* = -V~; 

)J]'=(-l)^ = + l; 

)i]«=(.-l)» = -l; 

)i]»=(-l)^ = + l; 

)J]» = (- 1)1 = (- i)^(- i)i = + ^—; and so 

)*]'=(- 1)5 =VPT)= -±-v/^orTl, 
oc?c? or eyen integer. Hence, 
The successive powers of \/~ 1 form the repeating series : 

+ V~h -h -V^» +1- 

The methods for operating with imaginary expressions are the 
same as those for surds ; but before applying the methods it is better 
to remove the factor /y/— 1. All cases of multiplication can be made 
a direct application of Arts. 97, 114. 



w- 


1? = 


= [(-1 


w- 


-xY- 


= [(-1 


w- 


~xY-. 


= [(-1 


w- 


-lY- 


= [(-1 


w- 


-^Y- 


= [(-1 


[V- 


-xY-- 


= [(-1 


[V- 


1]' = 


- [(-1 


w- 


T]' = 


= [(-1 


w- 


T/ = 


= L(-i 


on. In 


general, 


W~ 


T]" = 


=[(-1 


accordin 


gas 


n is an 



RADICAL EXPRESSIONS. 287 



niustratioiiB. ^/- 6aH^= y/S a^b* X (-1) = \/S aH^ X \/-l 
= 2ab \/Tb X V^. 



V-9«^ + V-49a«-V4a'«= 3a ^- 1 +7a/v/- 1 -2 a 
= 10a V^- 2a 
= 2 a (5 V=n - 1). 

3 >v/^ X 4 -v^^ = (3 V^ X \/-^)("* V^ X V-^) 
= 3 ^3 X 4 y 2 X ^/^ X V^ 
= 12V3X^X [(-l)*]' 

= - 12 ye. 

2 -v/^ X 5/v/^ X 3 V^ 
= 2 V3 X 5 V2 X 3 >v/6 X \/^ X ^^^ X V"^ 
= 30>v/3 X 2 X 6 X [(- l)*]" 
= - 180 ^~l. 

= f -v/3 X 1 = J V^- 

_ (i + y-H)' _ i+.2v^_+ (-0 

Example. Multiply 1-2 V"^ by 3 + ^y/^. 
ProcesB. 1—2 ^—~i 

3+ V-1^ 



1 - 2 -y/- 1 multiplied by 3, 3-6 V" ^ 

1-2 y'lH! multiplied by y'^, 2 + V^ 



Sum of the partial products, 6 — 5 /y/— I 



288 ELEMENTS OP ALGEBRA. 

Notes : 1. Imaginary expressions represent impossible'operations ; yet it is 
a mistalie to suppose that they are unreal, or that they have no importance. 

2, If the student employ the method of multiplying or dividing the expres- 
sions under the radicals (Arts. Ill, 113), for all cases in multiplication and divi- 
sion, he cannot readily determine the sign of the product or dividend. Thus, 



V^^ xV—a = V—aX—a= Va^ = ±a. 

3. Is the above product both ±a or — a ? We are limited to the considera- 
tion of the product of two equal factors, and we know that the sign of each is 
negative ; also, that Va^ = it «. Hence, the sign of Va^ will necessarily be 
the same as that of each of these factors. Therefore, it will be the same as was 

its root. Thus, 

V- 3 X 1/- 3 = - 1/9 = - 3, 



Exercise 103. 

Simplify : 

1. V^; '^-16; V- 12 a; V^^T^; V^ 

2. V-49a2-&6. ^^7729; ^IT^; y'^^^". 
Find the values of : 

3. (V— i)i^,(V^f ; iV-if; {-V—lf. 

4. i-V^lf; (-V=^r; i-V=-lf; {-V^f 



5. A/-25 - A/-49 + V-121 - a/-64 + V-1- a/-36. 



V-22 V-216 



6. 2 V- 24 + —= - V- 18 ; ^...^ - 



V-3 A/-33 V-324 



7. V- 36 a^ 4- V- 9 a^ - V- (1 - af a^ - V- a\ 



8. V-{ct-hf+ V-(a2- 2ab + b^)+ V-1 6 a^ b^-V- 4 a^ 
Multiply : 

9. V^ by V^; 3 V^ + V^^ by 4 V^^ 



RADICAL EXPRESSIONS. 289 

10. 2 V^ by 4 V'^; 1 + V^ by 1 + V^. 

11. V- 2 + 3 V^ by V^^ + 3 V^. 

12. 3-2 V-4 by 5 + 3 V^^; 4 + V^ by 4- V^. 

13. 1 + V^ by 1- V-1; 2 - V=^ by 1 - 2 V^~3. 

14. 2 V^ - 6 V^ by V^ + V^. 



15. Va — ^ by V^ — a ; a + V— a; by a — V— a;. 

16. a V— a + b V— b by a V— a — 5 V— 6. 
Divide : 

17. V^^ by V^^; - \/^ by - 6 V^. 

18. V^ by V- 20; V- 24 - V^ by a/^^ 



19. 2 V— 4 «*-» by V— a^ \ a + V- a by V- a^. 

20. - 2 V^ by 1 - \/^ ; 2 by 1 + V^. 

21. \^-^^^ by v^- 5; '^^^^ by ^=^. 

22. 4 + V^ l)y 2 - V^; V^3 by 1 - V^. 
Rationalize the denominators of : 

23 ^i^J^^^- 2 \/:ri _ 3 yry 3 + 3 V^ 

" ' 2 - V^' 4 a/^ + 5 V^^' 2-2 V=I ' 

Queries. To what form can all imaginary monomials be reduced ? 
In multiplication and divi.sion why separate the imaginary expres- 
sions into their sunl and imaginary factors ? Is it necessary in all 
? 

19 



290 ELEMENTS OF ALGEBRA. 

Quadratic Surds. 

116. I. A quadratic surd cannot equal the sum or differ- 
ence of a rational expression and a quadratic surd. 

Proof. If possible, let ^/a = 6 db ^\fc, in which ^a and y^c 
axe dissimilar quadratic surds, and 6 a rational expression. 

Square both members, a = 6"^ ± 2 & ^/c + c. 

± a T ^^ T c 



Transpose, ± 2 6 \/c = a — ft^ — c*. .♦. ^/c 



26 



That is, a surd equal to a rational expression, which is impossible. 
Therefore, ^\/a cannot equal h ± ^ c. 

II. i/" a + Vb = X + Vy, in which a and x are rational 
and Vb and Vy cire quadratic surds, prove that a = x and 
b = y. 

Proof. Transposing, /y/6 = (x — a) + V^?/- Now if a and a; were 
unequal, we would have a quadratic surd equal to the sum of a ra- 
tional expression and a quadratic surd, which, by L, is impossible. 
Hence, a = x. Therefore, ^Jh = ^^y, ot b = y. 

III. 7/" V a + Vb = Vx + Vy, prove that y a — Vb 
= Vx — vV- 

Proof. Square both members, a + \/b = x + 2 aJx y + y. 
Therefore IL, a = x + y {!) and a/6 = 2 ^/x^ (2) 

Subtract (2) from (1), a - \/b = x ~ 2 \/xy + y. 
Extract the square root, V a — \/b = y\/x — \/y. 
Similarly it may be shown that if V « — ^/b = ^/x — /y/jr, 

then V a + ^Jb = ^Jx + ^/y. 



RADICAL EXPRESSIONS. 291 

Square Root of a Quadratic Surd. 
117. To find the square root of a binomial surd a ± Vh. 



Process. Let Va ± a/6 = V^ ± Vi^ (1) 

Then (III., Art. 116), Va T V^ = V-^ T \/y (2) 

Multiply (1) and (2) together, ^af^ b = x- y (3) 

Square (1), a ± aA = x ±2 ^x y + y. 
Therefore (II., Art. 116), a=x-hy (4) 

Add (3) and (4), a + v^^"^ = 2 x. .-. x = ^ 

, a - Va* — b 

Subtract (3) from (4), a - V a* -b = 2y. .'. y = ^ 

Therefore, V7±Vb = \J ^^ V^^ ± \/" ~ ^"'~^ (0 

HotM; 1. Evidently, unless d^ — h be a perfect square, the values of Vx 
and Vy will be com plex surds ; and the expression Vx + \ y will not be as 
simple as V a + 1/6. 

2. Since, Va'^c + Vbc — \/c{a ^- Vh\ also if a^ - 6 be a perfect square 
the squats root of a + Vh may be expressed in the form Vx 4- Vy, the square 
root of Vcflc ■{■ Vbc'is of the form \'c ( V'x -f Vy ). 

3. Frequently the square root of a binomial surd may be found by in- 
si>ection. Thus, 

FiTid ttoo numbers whose sum is the rational term^ and whose product is the 
square of half the radical term. Connect the square roots of these numbers by 
the sign of the radical term. 

Examples : 1. Find the square root of 3^ - VlO. 
Process. Let Vi - Vy = "^H - aAo (1) 

Then (III., Art. 116), y'i -f Vy = V^T'^ (2) 

Multiply (1) and (2) together, x-y= \/*^ _ lo = | (3) 

Square (1), x - 2 V^ 4- y = 3i - - /y/lO^ 

Therefore (II., Art. 116), z + y = 3| (4) 

From (.3) and (4), x = 2|, y=l. 

Therefore, ^3^ - \/Tb = a/| -I = ^ V^- 1- 



292 ELEMENTS OF ALGEBRA. 

We may employ the general form (i). Thus (since a = 83 and 
y6 = + 12 V35), 

L^ ^ ^g^2 _ (|2 ^35)2 

2. V83 + 12 ^/3b = \ ^^ — - 

4/ 83 - V8'3^ - (12 V35f _ / 83 + 43 / 83 - 43 

= ^63 + ^20 = 3 v^ + 2 Vs. 



3. Va/ST - 2 a/6 = V V3 (3 - 2 a/2) rr: ^3 X a/3 - 2 a/2, ^ 



also \/3 - 2 a/2 (in which a = 3 and a/& = - 2 a/2) 

^ i / 3 + V3-^ - (2 A/2y _ i / 3 - V3"^ (2 a/2)^ ^ ^ _ i 



.-. a/ a/27 - 2 Ve = a!^3 (a/2 - l). 

4. Find by inspection the square root of 103 — 12 a/h. 

Solution. The two numbers whose sum is 103 and whose 
product is (^ — |— j , are 99 and 4. Hence, Vl03 - 12 \/Tl 
=: a/99 - a/4 = 3 a/iT - 2. 



5. Similarly, VlO + 2 a/21 := a/7 + Vs, because 7 an^ 3 are 
the only numbers whose sum is 10 and whose product is (a/21) • ^ ^ 



Exercise 104. 

Find the square roots of: 

1. 7-2A/rO; 5 + 2V6; 41-241/2; 2J + V^. 

2. 18-8 V5 ; 11 + 2 VSO ; 13 - 2 a/42. 

3. 15~V56; 47 -4 a/33; 6-2^5; 10 + 4 a/6. 



RADICAL EXPRESSIONS. 293 



5. V27 + a/15 ; 2?7i + 1 + 2 Vm^ + 7i - 2. 

6. (m^ + m) ?i - 2 vi n Vm ; 9 — 2 VU. 

7. (wi + w)'-^ — 4 (m - /O A/m?i ; 3 a; - 2 a: V2. 

Find the fourth roots of: 

8. 97-56V3; | a/5 + 3J ; 56 + 24 V5. 

9. 17 + 12V2; 4(31 -8 Via); 248 + 32 V^. 

Simple Equations Containing Surds. 



118. Examples : 1. Solve V4 ar^ - 7 x + 1 = 2 a; - 4 (1) 

Process. Square (1), 4x^-7x+l = 4x^-7\x + ^. 
.'. X = ]l\. Hence, 

To Solve an Equation containing a Single Surd. Arrange 
the terina bo as to have the surd alone in one member, and then raise 
each member to the power indicated by the root index. 

Note. If the equation contains two or more surds, two or more operations 
may be necessary in order to clear it of radicals. Thus, 



2. Solve \/'2b x-1%- ^4x- 11=3 -^/i. 

Process. Transpose, \/25a:-29 = 3 ^/x 4- *J\x-\\ (1) 

Square (1), 25x-29 = 9x+6 V(4a:- ll)a:+4a:-ll. 

Transpose, etc., 'v/(4a;- 11) x = 2 x — 3 (ii) 

Square(2), 4x«- llz = 4a;«-12ar + 9. .-. x = 9. 



294 ELEMENTS OF ALGEBRA. 

3. Solve ^^-7= = -^ (I) 

\ X + n ^x + 3n ^ ' 

Process. Clear (1) of fractions, transpose and unite, etc., 

/ ^ r TT /- mn ( mn \2 
(m — n) \/x = mn. Hence, \/x = . .*. x = 



^ . \/m + X 4- ym — x 

4. feolve ^' , J — n. 

\m -\- X — 's/m — X 

Process. Rationalize the denominator, 

fn. + 's/m'^ — x'' 



(1) 



From (1), y'rn^ - x"^ - nx - m (2) 

Square (2), m^ ~ x^ = n^ x^ -"Imnx ■\- m\ 

Transpose, etc., a;^ (l + n^) =r 2 mna:. 

Divide bv a;, a:(l + w^) =: 2mn. .*. a: i= 7— — ;, 

Exercise 106. 

Solve : 



1. V^ + 5 = 4 ; V3 ^^ + 6 = 6 ; ^x^-2^x^% 

2. Va:2 _ 3 ^. _}_ 5 ^ :i; _ 1 ; ^2^-3 -1=2. 

3. V'3 + V4 + V^^^= 2; Vr+W^^m = a: + 2. 

4. ^/mx^~a = 'v/c^TT ; V^rr2 + ^4 - ^3^ = 2:. 

5. VJT~2 =. 2 V'2ic-3 ; V3 a: + 5 = 3 ^"Ix ^ 1. 



6. V3.r- 4= V2a: + 16; ^^2 ^' - 4 = V 4 - V2 rr. 



7. 3Vi = 



RADICAL EXPRESSIONS. 
8 



295 



V9 a; - 32 



4- V 9 a; - 32. 



8. Va: + 3 4- Vx + S - V4 :c + 21 = 0. 

9. ^Vx+ - ^Vx — 5 = y/2 Vi. 

Q 

10. 'V^m^ -f a; Vn^ + x^ = Vx-{- vi; Vx + Vx—2 = —r' 

\ X 

11. A^4+2V2^-5 = V3; V| + :^^m_ 

V a: — V m ^ 



V2 ic + 1 + 3 Va; _ Vm a; — n __ 3 v m a; — 2 7i 
a/2 a; + 1 — 3 Va; Vwi x-\- n 3 Vw x + 5n 



13 



V5a;+ VB Va;4- 5 V6a;+2 _ 4:V6x + 6 
V3^ 4- V3 ~ V^ + 3' V6^- 2 ~ 4 V6'^ - 9 

14. v/^+v/^^=c/4^- 

"w + a; 'm — a; »7?r — ar 
Solve the following for x and y : 

15. a; + 4 V3 + y = 15 — a; 4- y V5. 

16. a; 4- y 4- a; Va 4- y V^ = 1 — Va. 

17. a: - 5 + (2 y - 3) V3 = 5 a; - Vl2. 

18. a; — rt 4- {y — 3) Va 4- 2> = w a; 4- Vol. 



19. a — Va; + y = y — x — Vm 4- 71. 

20. a; Vw(Vm4- 1) = w — wi4- y V^(l — V^. 



296 ELEMENTS OF ALGEBRA. 



CHAPTEK XX. 
LOGARITHMS. 

119. If a^ = m, then / is called the logarithm of m to the 
base a. Hence, 

A Logarithm is the exponent by which a certain num- 
ber, called the base, must be affected in order to produce a 
given number. 

The logarithm of m to the base a is written logam. Thus, 
log„m = I expresses the relation a^ = m; logj^ 100 = 2 expresses the 
relation 10^ = 100, etc. 

Since numbers are formed by combinations of tens, any number 
may be expressed, exactly or approximately, as a power of 10. Thus, 



1000 = 103 . ei-c 

120. Common System of Logarithms. This system has 
10 for its base, and is the only one used for practical 
calculations. Thus, 

Since 100=1, log 1 = 0; since 10^ = 10, log 10 = 1 ; 

since 102 = iqO, log 100 = 2 ; since lO^ = 1000, log 1000 =: 3; 

since 10* = 10000, log 10000 = 4 ; and so on. 

Since lO-i = J^ = .1, log .1 = - 1 = 9 - 10 ; 
since 10- 2 = ^i^ = .01, log .01 = - 2 = 8 - 10 ; 
since lO-s = ^Jq^ = .001, log .001 = - 3 = 7 - 10 ; and so on. 

It is evident that the logarithm of all numbers greater than 1 is 
positive J and of all numbers between and 1 is negative ; also, that 
the logarithm of any numbers between 



LOGARITHMS. 297 

1 and W is -f a fraction; 

10 and 100 is 1 + a fraction ; 
100 aiid 1000 is 2 + a fraction ; 

1 and .1 is — 1 -f a fraction, or 9 + a fraction — 10; 
.1 and .01 is — 2 + a fraction, or 8 + a fraction — 10; 
.01 and .001 is — 3 + a fraction, or 7 + a fraction — 10; and so on. 

It thus appears that the logarithm of a number consists 
of an integral part, called the characteristic, and a I'ractional 
part, called the mantissa. 

The mantissa is always made positive. 

IUu8tration8. It is known that log 5 = 0.69897; log 12 = 1.07918; 
log 2912 = 3.4(3419; etc. These results mean that loo«»8»' = 5; 

1O1.07918 ^ 12; 108.4M19 = £912 ; CtC. 

Notes : 1 . 'Hie fractional part of a logarithm cannot be expressed exactly, 
but an apiiroxiniate value may be found, true to as many decimal places as 
desire^l. Thus, the logarithm of 3 is found to be 0.477121, true to the sixth 
place. 

2. For brevity the expression "logarithm of 3" is written log 3. The 
expression "log «" is read "logarithm of x." 

3. Logarithms were inventetl by John Napier, Baron of Merchiston, Scot- 
land, and first published in 1614. 

4. Ih-re are only two systems of logarithms in general use : the Natural, 
or Hyperbolic, system, and the lirhjtjsian, or Common, system. The base sub- 
script of the former is e, and that of the latter is 10. 

5. The natunil system, invente<l by Jolm Speidell and published in 1619, is 
employetl in the higher branches of analysis and in scientific investigations ; 
its base is 2.718281828+ . 

6. The common system, more properly called the denary or decimal sys- 
tem, was invente<i by Henry Brigps, an Engli.sh geometrician, and first pub- 
lished in 1617. The logarithm of its base, 10, is alrmys 1. 

7. The logarithms invente<l by Napier are entirely different from those in- 
ventetl by Si>eidell, though they are closely connected with them. The natural 
system may be regardeil as a modification of the original Napierian system. 

121. Since log 1 = 0, log 10 = 1, log 100 = 2, log 1000 = 3, etc., 
the characteristic of the logarithms of all numbers consisting of one 



298 ELEMENTS OF ALGEBRA. 

integral digit (that is, all numbers with one figure to the left of its 
decimal point) is 0; of all numbers consisting of two integral digits 
is 1 ; of all numbers consisting of three integral digits is 2 ; and so 
on. Hence, 

I. The characteristic of the logarithm of an integral 
number, or of a mixed decimal, is one less than the numher 
of integral places. 

Since log .1 = - 1, log .01 = - 2, log .001 = - 3, etc. ; the charac- 
teristic of the logarithm of any decimal whose first significant figure 
occupies the first decimal place (that is, of any number between 0.1 
and 1) is — 1 ; of any decimal whose first significant figure occupies 
the second decimal place (that is, of any number between 0.01 and 
0.1) is — 2 ; of any decimal Vfho^Q first significant figure occupies the 
third decimal place (that is, of any number between 0.001 and 0.01) 
is — 3; and so on. Hence, 

II. The characteristic of the loga7nthm of a decimal is 
negative, and is numerically equal to the numher of the place 
occupied hy the first aignificant figure of the decimal. 

The characteristic only is negative. Hence, in the case of decimals 
whose logarithms are negative, the logarithm is made to consist of a 
negative characteristic and a positive mantissa. To indicate this, the 
minus sign is written over the characteristic, or else 10 is added to 
the characteristic and the subtraction of 10 from the logarithm is 
indicated. 

Thus, log .0012 = 3.0792, or 7.0792 - 10 ; read "characteristic 
minus three, mantissa nought seven ninety-two," or "characteristic 
seven minus ten, etc." In reading the mantissa, for brevity, two inte»- 
gers are read at a time. Thus, log 2 = 0.30103, is read "the loga- 
rithm of two equals characteristic zero, mantissa thirty ten three." 

Illustrations. The characteristic of the logarithm of 9 is 0; 
of 32 is 1 ; of 433 is 2 ; of 39562 is 4; of 632.526 is 2 ; of .42 is - 1 ; 
of .023622 is - 2 ; of .0000325 is - 5 ; etc. 



LOGARITHMS. 299 

122. Let m and n be any two numbers whose logarithms are x 
and y in the common system. Then l(F = in and 10*= n. Multiply- 
ing the equations together, we have 1(F+*' = mn. Hence (Art. 1 19), 
log mn = z + y. But x = log m and y = log n. Therefore, log m n 
= log m -H log n. Similarly log in np = log m -f log n + log p ; etc. 
Hence, 

Tlie logarithm of a product is found by adding together 
Die logarithms of its factors. 

Ulustrationa. Given l(»g 2 = 0.3010, log 3 = 0.4771, log 5 
= 0.6990, log 7 = 0.8451. 

log 252 = log (2 X2X3X3X7) 

= log 2 + log 2 + log 3 -f log 3 + log 7 

= 2 log 2 + 2 log 3 + log 7 

= 2 X 0.3010 + 2 X 0.4771 + 0.8451 

= 0.6020 + 0.9542 + 0.8451 

= 2.4013. 

log 300 = log (2 X 3 X 5 X 10) 

= log 2 + log 3 + log 5 + log 10 
= 0.3010 + 0.4771 + 0.6990 + 1 
t= 2.4771. 

Exercise 106. 

Given log 2 = 0.3010, log 3 = 0.4771, log 5 = 0.6990, 
log 7 = 0.8451 ; find the values of the following : 

1. log 6; log 64; log 14; log 8; log 12; log 15; log 84. 

2. log 343; log 16; log 216; log 27; log 45; log 36. 

3. log 90; log 210; log 3600; log 1120; log 1680. 

123. If any number be multiplied or divided by any integral 
power of 10, since the sequence of the digits in the resulting number 
remains the same^ the mantis-sa) of their logarithms will be unaffected. 
Thus, since it is known that log 577.932 = 2.7619, 



300 ELEMENTS OF ALGEBRA. 

log 5779.32 = log (577.932 X 10) = log 577.932 + log 10 

2.7619 + 1 = 3.7619. 

log 57793.2 = log (577.932 X 100) = log 577.932 + log 100 

2.7619 + 2 = 4.7619. 

log 57.7932 = log (577.932 X 0.1) = log 577.932 + log 0.1 

2.7619 + (- 1) rr 1.7619. 
log 5.77932 = log (577.932 X 0.01) = log 577.932 + log 0.01 

2.7619+ (-2) = 0.7619. 

log .577932 = log (577.932 X 0.001) = log 577.932 + log 0.001 

= 2.7619 + (-3) =1.7619. 
Etc. Hence, 

The mantissce of the logarithms of numbers having the 
same sequence of digits are the same. 

Illustrations. If log 44.068 = 1.6441, log 4.4068 = 0.6441, 
log .44068 = 1.6441 or 9.6441 - 10, log .000044068 = 5.6441 or 
5.6441 - 10, log 440.68 = 2.6441, log 440^800 = 6.6441, etc. If 
log 2 = 0.3010, log .2 = 1.3010, log .02 ^ 2.3010, log 20 = 1.3010, 
etc. Hence, 

The mantissa depends only on the sequence of digits, and 
the characteristic on the position of the decimal point. 



Exercise 107. 

1. Write the characteristics of the logarithms of : 12753; 
13.2; 532; .053; .2; .37; .00578; .000000735; 1.23041. 

2. The mantissa of log 6732 is .8281, write the logarithms 
of: 6.732; 673.2; 67.32; .6732; .006732; .000006732. . 

3. Name the number of digits in the integral part of the 
numbers whose logarithms are: 5.3010; 0.6990; 3.4771. 



LOGARITHMS. 301 

4. Name the place occupied by the first significant fig- 
ure iu the numbers whose logarithms are: 4.8451; 0.7782. 

Given log 2 = 0.3010, log 3 = 0.4771, log 5 = 0.G990, 
log 7 = 0.8451; find the logarithms of the following 
number : 

5. .18; 22.5; 1.05; 3.75; 10.5; 6.3; .0125; 420. 

6. .0056; .128; 14.4; 1.25; 12.5; .05; .0000315. 

7. .3024; 5.4; .006; .0021; 3.5; .00035; 4.48. 

124. Let 7« be any number whose logarithm is x. Then UF^wi. 
Raising both members to the pih power, we have 10'" = inP. Hence 
(Art. 119), log mf = px. But x — log?n. Therefore, log Tn** 
= -p log m. Similarly log wi^'n' = p log m -\- q log n, etc. Hence, 

Tlie logarithm of any power of a number is found h/ 
multiplying the logarithm of tlie number by the exponent of 
Vie power. 

niustrations. log 5" = 10 log 5 =J0 X 0.6990 = 6.99(H), 
log .003* = 5 log .(K)3 = 5 X 3.4771 = 13.3855. 
log 864 = log 26 X 3« = 5 log 2 + 3 log 3 = 5 X 0.3010 + 3X0 4771 
= 1.5060. 

Note. If the number i.s a decimal and the exponent positive, the j)roduct of 
the characteristic and exponent will be negative, and since the mantissa is made 
positive, we must algebraically add whatever is carried from the niantis.sa. 

Thii.«», log .0005" = 10 X 4.6990 = 40 + G.9900 = 34 + 0.9900 = 34.9900. 

Exercise 108. 

Given log 2 = 0.3010, log 3 = 0.4771, log 5 = 0.6990, 
log 7 = 0.8451 ; find the logarithms of: 

1. 2*; 53; 7^; 8^ 3^; 64; 81; 72; (8.1)7; (2.10)6 

2. 343; .036; .000128; (.0336)1^ (.00174)2; (3.84)» 



302 ELEMENTS OF ALGEBRA. 

125. Let m and n be any two numbers whose logarithms are x 
andy. Then 10^ = m and 10^' = w. .-. 10^-2' = m -f n. 

m " mn 

log - = a: - 2/ = log m - log n. Similarly, log ^^^ = log m + log n 

— (log wij + log Uj). Etc. Hence, 

The logarithm of a quotient is found by subtracting the 
logarithm of the divisor from the logarithm of the dividend. 

Illustrations, log | = log 3 -log 2 = 0.4771-0.3010 = 0.1761. 
log f = log 5 - log 7 = (0.6990) - 0.8451 = (1.6990 - 1) - 0.8451 
= 0.8539 - 1 = 1.8539. 

Note. To subtract a greater logarithm from a less logarithm. Add to the 
characteristic of the minuend the least number which will make the minuend 
greater than the subtrahend ; also indicate the subtraction of the same number 
from the minuend so increased. Then proceed as before. Thus, 

log =^ = log 252 - log 300 = (2.4014) - 2. 4771 = (3.4014-1) - 2. 4771 = 1.9243. 

log '-^ = (3.6990) - 2T8451 = (T.6990 - 2) - 2.8451 = 0.8539-2 = 2.8539 or 
8.8539 - 10. 

Exercise 109. 

Given log 2 = 0.3010, log 3 = 0.4771, log 5 = 0.6990, 
log 7 = 0.8451 ; find the logaritlims of: 



1 ^ 

2' 



_3^ 7 3 .003 .005 /SV .007 

' .05' 5' 5' 2' Q7 ; 02 ' VlO/ ' -02' 



2 42. -:!!_. 125- '^- ^- ^i^. 5. ^ 
■ ^' .0052' ' 5 ' 8.1' .000027' ' .007* 

126. Let m be any number of which the logarithm is x. Then 

X 

1(F = m. Taking the rth root of each member, we have 10'" = \/m. 
.'. (Art. 119), log \/m — - = -^ — . Similarly, log ^^m n 



LOGARITHMS. 303 

The logarithm of any root of a numher is found hy divid- 
ing the logarithm of the number by tJie index of the root. 

«/- log 5 0.6990 ^ _„ 
IHuatrations. log ^ = -|- = — 5— = 0.1398. 

,, log.0(X)7 4.8451 3.8451 + 7 ^„ - -,,^« 

log '^/ioOO? = -^ = -y— = 7-^ = 0.5493+1 = 1.5493. 

1 VT-^. logl5<^ 5 log 3 X .5 5 (log 3 + log .5) 
logVl-5»=— ^r— = e = 6 

^5(0.4771 + 1.6990)^^^^^^^ 
o 

Note. If a negative characteristic is not exactly divisilile by tlie index of 
the root, subtract from the characteristic the least positive number which will 
make it so divisible. Indicate the addition of the characteristic so formed to 
the mantissa, and prefix the number subtracted from the characteristic to the 
mantissa. Then divide separately. Thus, 

log V75 = ?5|:5 = I:^ = '•6»y + ^ = 0.8495 + T = T.8495 or 9.8495 - 10. 

Ik ii it 



Exercise 110. 

Given log 2 = 0.3010, log 3 = 0.4771, log 5 = 0.6990, 
log 7 = 0.8451 ; find the logarithms of: 

1. ^7; \^l; \/2- v^.^; ^243; ^12^; \^M; ^^. 

2. ^^iTK2f; 5ix3i; ^,; g; ^^^; 6i x 3f. 

,, ^^x\/2 j^^ ^ 7/ J.21)2_ V^2^ 
"• -;/l8xV2^ ^15' -^!^7' V(.()()084)2' ^^^^^fg ' 

127. Table of Logarithms. The table (pages 304 and 305) 
gives the nianti.Hsae of the logarithms to four decimal i>]jices for all 
numbers from 1 to 10(X) inclusire. The characteristic and decimal 
points are omiUedy and must be supplied by inspection (Art. 121. 



304 



ELEMENTS OF ALGEBRA. 



N 


1 


2 


3 


4 6 


6 


7 


8 


9 


10 

11 
12 
13 
14 


ouoo 

0414 
0792 
1139 
1461 


0043 
0453 
0828 
1173 
1492 


0086 
0492 
0864 
1206 
1523 


0128 
0581 
0899 
1239 
1558 


0170 
0569 
0934 
1271 
1584 


0212 
0607 
0969 
1803 
1614 


0253 
0645 
1004 
1335 
1644 


0294 
0682 
1038 
1367 
1673 


0384 
0719 
1072 
1899 
1703 


0874 
0755 
1106 
1480 
1732 


15 

16 
17 
18 
19 


1761 
2011 
2304 
2553 

2788 


1790 
2068 
2330 

2577 
2810 


1818 
2095 
2355 
2(501 
2833 


1847 
2122 
2380 
2625 

2856 


1875 
2148 
2405 
2648 

2878 


1908 
2175 
2430 

2672 
2900 


1931 
2201 
2455 

2695 
2928 


1959 

2227 
2480 
2718 
2945 


1987 
2253 
2504 
2742 

2967 


2014 
2279 
2529 
2765 
2989 


20 

21 
22 
23 
24 


3010 
8222 
8424 
3617 

3802 


;;o32 

3243 
3444 
3636 
3820 


3054 
8263 
8464 
3655 

3838 


3075 
3284 
3488 
3674 
3856 


3096 
3304 
8502 
3692 

3874 


3118 
3324 
3522 
8711 
3892 


3139 
3845 
8541 
3729 
3909 


3160 
3365 
3560 
3747 
3927 


3181 
3385 
3579 
3766 
3945 


8201 
3404 

3598 
3784 
3962 


25 

26 
27 
28 
29 


3979 
4150 
4814 
4472 
4624 


8997 
4166 
4330 

4487 
4639 


4014 
4183 
4346 
4502 
4654 


4031 
4200 
4362 
4518 
4669 


4048 
4216 
4378 
4533 
4683 


4065 
4282 
4393 
4548 
4698 


4082 
4249 
4409 
4564 
4713 

4857 
4997 
5132 
5263 
5891 


4099 
4265 
4425 
4579 

4728 


4116 
4281 
4440 
4594 
4742 


4133 
4298 
4456 
4609 
4757 


30 

31 
32 
33 
34 


4771 
4914 
5051 
5185 
5315 


4786 
4928 
5065 
5198 
5328 


4800 
4942 
5079 
5211 
5340 


4814 
4955 
5092 
5224 
5353 


4829 
4969 
5105 
5237 
5866 


4843 
4983 
5119 
5250 

5378 


4871 
5011 
5145 
5276 
5403 


4886 
5024 
5159 
5289 
5416 


4900 
5038 
5172 
5302 
5428 


35 

36 
37 
38 
39 


5441 
5568 

5682 
£798 
5911 


5453 
5575 
5694 
5809 
5922 


5465 
5587 
5705 
5821 
5933 


5478 
5599 
5717 
5882 
5944 


5490 
5611 
5729 
5848 
5955 


5502 
5623 
5740 
5855 
5966 


5514 

5635 
5752 
5866 
5977 


5527 
5647 
5763 

5877 
5988 


5539 
5658 
5775 
5888 
5999 


5551 
5670 
5786 
5899 
6010 


40 

41 
42 
43 
44 


6021 
6128 
6282 
6335 
6435 


6031 
0138 
6243 
6345 
6444 


6042 
6149 
6253 
6355 
6454 


6053 
6160 
6263 
6365 
6464 


6064 
6170 
6274 
6375 
6474 


6075 
6180 
6284 
6385 
6484 


6085 
6191 
6294 
6395 
6493 


6096 
6201 
6304 
6405 
6508 


6107 
6212 
6314 
6415 
6513 


6117 
6222 
6825 
6425 
6522 


45 

46 
47 
48 
49 


6532 
6628 
6721 
6812 
6902 


6542 
6637 
6730 
6821 
6911 


6551 
6646 
6789 
6830 
6920 


6561 
6656 
6749 
6889 
6928 


6571 
6665 
6768 
6848 
6937 


6580 
6675 
6767 
6857 
6946 


6590 
6684 
6776 
6866 
6955 


6599 
6693 
6785 
6875 
6964 


6609 
6702 
6794 
6884 
6972 


6618 
6712 
6803 
6893 
6981 


50 

51 
52 
53 
64 


6990 
7076 
7160 
7243 
7324 


6998 
7081 
7168 
7251 
7382 


7007 
7093 
7177 
7259 
7340 


7016 
7101 

7185 
7267 
7348 


7024 
7110 
7198 
7275 
7356 


7033 
7118 
7202 
7284 
7364 


7042 
7126 
7210 
7292 
7372 


7050 
7135 
7218 
7300 
7380 


7059 
7148 

7226 
7308 
7388 


7067 
7152 
7285 
7316 
7396 



LOGARITHMS. 



305 



N 





1 


2 


3 


4 


6 


6 


7 


8 


9 


55 

56 
67 
68 
69 


7404 
7482 
7559 
lOU 
7709 


7412 
7490 
7566 
7642 
7716 


7419 
7497 
7574 
7(W9 
7723 


7427 
7505 
7582 
7657 
7731 


7435 
7513 
7689 
7664 
7738 


7448 
7520 
7597 
7672 
7745 


7451 
7528 
7004 
7079 
7752 


7459 
7636 
7612 
7686 
7760 


7466 
7643 
7610 
7694 
7767 


7474 
7661 
7627 
7701 

7774 


00 

61 
62 
63 
64 


7782 
7853 
7924 
7993 
8062 


7789 
7800 
7931 
8000 
8069 


7796 
7868 
7938 
8007 
8075 


7803 
7875 
7945 
8014 
8082 


7810 
7882 
7952 
8021 
8089 


7818 
7889 
7959 
8028 
8096 


7825 
7896 
7966 
8035 
8102 


7832 
7903 
7973 
8041 
8109 


7889 
7910 
7980 
8048 
8116 


7846 
7917 
7987 
8056 
8122 


65 

66 
67 
68 
69 


8129 
8195 
8261 
8325 
8388 


8136 
8202 
8267 
8331 
8895 


8142 
8209 
8274 
8338 
8401 


8149 
8215 
8280 
8344 
8407 


8156 
8222 
8287 
8351 
8414 


8162 
8228 
8293 
8357 
8420 


8169 
8236 
8299 
8363 
8426 


8176 
8241 
8306 
8370 
8432 


8182 
8248 
8812 
8376 
8430 


8189 
8264 
8819 
8382 
8446 


70 
71 
72 
78 

74 


8451 
8513 
8573 
8633 
8692 


8457 
8519 
8579 
8639 
8098 


8463 
8525 
8586 
8645 
8704 


8470 
8531 
8691 
8651 
8710 

8768 
8825 
8882 
8938 
8993 


8476 
8637 
8697 
8657 
8716 


8482 
8643 
8603 
8663 
8722 


8488 
8549 
8609 
8669 
8727 


8494 
8555 
8015 
8675 
8733 


8600 
8661 
8621 
8681 
8739 


8606 
8667 
8627 
8686 
8746 


75 

76 

77 
78 
79 


8761 

8808 
8866 
8921 
8976 


8756 
8814 
8871 
8927 
8982 


8702 
8820 
8876 
8932 
8987 


8774 
8831 
8887 
8943 
8998 


8779 
8837 
8893 
8949 
9004 


8786 
8842 
8899 
8954 
9009 


8791 
8848 
8904 
8960 
9015 


8797 
8864 
8910 
8966 
9020 


8802 
8869 
8916 
8971 
0025 


80 
81 
82 
83 
84 


9031 
9086 
9138 
9191 
9243 


9036 
9090 
9143 
9196 
9248 


9042 
9096 
9149 
9201 
9263 


9047 
9101 
9164 
9206 
9268 


9053 
9100 
9150 
9212 
9263 


9058 
9112 
9106 
9217 
9269 


9063 
9117 
9170 
9222 
9274 


9069 
9122 
9176 
9227 
9279 


9074 
9128 
9180 
9232 
9284 


9079 
9183 
0186 
0238 
9289 


85 
86 
87 
88 
89 


9294 
9346 
9396 
9446 
9494 


9299 
9360 
9400 
9460 
9499 


9304 
9355 
9405 
9456 
9604 


9309 
9360 
9410 
9460 
9600 


9316 
9365 
9416 
9465 
9613 


9320 
9370 
9420 
9469 
9618 


9325 
9375 
9426 
9474 
9628 


9330 
9380 
94:^ 
9479 
9528 


9836 
0386 
0435 
9484 
9533 


9840 
9390 
9440 
9489 
9638 


90 
91 
92 
93 
94 


9642 
9690 
9688 
9686 
9731 


9647 
9696 
9643 
9689 
9786 


9652 
96C0 
9647 
9694 
9741 


9667 
9606 
9662 
9699 
9746 


9662 
9609 
9667 
9703 
9760 


9666 
9614 
9661 
9708 
9764 


9571 
9019 
9666 
9713 
9759 


9576 
9624 
9671 
9717 
9763 


9581 
9628 
0675 
9722 
9708 


9686 
9633 
9680 
9727 
9773 


95 

96 
97 
98 
99 


9777 
9823 
9868 
9912 
9966 


9782 
9827 
9872 
9917 
9961 


9786 
9832 
9877 
9921 
9966 


9791 
98:^6 
9881 
9926 
9969 


9796 
9841 
0886 
0930 
9074 


0800 
9846 
0890 
9984 
9078 


9805 
9850 
9894 

9083 


0809 
9864 
0809 
0943 
0987 


0814 
9859 
91K)3 
9948 
99*^1 


9818 
0863 
9908 
0962 
0906 



20 



a06 ELEMENTS OF ALGEBRA. 

Explanation of Table. The left-hand column, headed N, is a 
column of numbers. The figures O, 1, 2, 3, 4, 5, 6, 7> 8, 9, 

opposite N at the top of the table, are the right-hand figures of num- 
bers whose left-hand figures are given in the column headed N. The 
figures in the column which they head are the corresponding man- 
tissse of the logarithms of the numbers. 

128. To Find the Logaritlmi of a Number. 

I. Consisting of one Figure. The mantissae of the logarithms 
of single digits, 1, 2, 3, 4, etc., are seen opposite 10, 20, 30, 40, etc., 
and in the column headed O. To the mantissa prefix the character- 
istic and insert the decimal point. Thus, 

log 6 = 0.7782. log .6 = 1.7782. log 8 = 0.9031. 

Similarly, since the mantissa of log .009 is the same as the man- 
tissa of log 9, log .009 - 3.9542. 

II. Consisting of t-wo Figures. In the column headed N look 
for the figures. In the line with the figures, and in the column 
headed 0, is seen the mantissa. Then proceed as before. Thus, 

log 13 =1.1139. log 2.5 = 0.3979. log .92 = 7.9638. 

Similarly, log .00092 = 4.9638. 

III. Consisting of three Figures. In the column headed N, 
look for the first two figures, and at the top of the table for the third 
figure. In the line with the first two figures, and in the column 
headed by the third figure, is seen the mantissa. Then proceed as 
before. Thus, 

• log 313 = 2.4955. log 17.9 = 1.2529. log .279 = T.4456. 
Similarly, log .000718 = 4.8561. 

IV. Consisting of more than three Figures, Take the man- 
tissa of the logarithm of the first three figures as given in the table. 
Prefix a decimal point to the remaining figures of the number, and 
multiply the result by the tabular * difference. Add the product to 

* The Tabular difference is the difference between the two successive raan- 
tissjfi between which the required, or given, mantissa Ues. 



LOGARITHMS. 307 

the mantissa thus taken. Prefix the characteristic and insert the 
decimal point as before. Thus, 

1. Find the logarithm of 80.672. 

The tabular mantissa of the logarithm of 806 is 9063 

The tabular mantissa of the logarithm of 807 is 9069 

Therefore, the tabular difference = 6 

The number 80672 being between 80600 and 80700, the mantissa 
of its logarithm must be between 9063 and 9069. An increase of 100 
in 80600 causes an increase of 6 in the mantissa of the logarithm of 
80600. Therefore, an increase of 72 in 80600 will produce an increase 
of ^ of 6 (or .72 X 6), or 4.32, in the mantissa of the logarithm of 
80600. Hence, the tabular mantissa of log 80672 mtist be 9063 + 4, 
or 9067. Prefixing the characteristic and inserting the decimal point, 
we have 

log 80.672 = 1.9067. 

Similarly, since the mantissa of log .0005102 is the same as the 
mantissa of log 5102, 

2 To find the logarithm of .0005102. 
The tabular mantissa of log 510 is 7076 

The tabular mantissa of log 511 is 7084 

.*. the tabular diflference = 8 

Hence, the tabular mantissa of log 6102 must be 7076 -f .2 X 8, or 
7078. 

.-. log .0005102 = 4.7078. 

Exercise 111. 

Find by means of the table the logarithms of the 
following : 

1. 70; 102; 201; 999; .712; 3.6; .00789; 3.21. 

2. .0031; .0983; .00003; 10.08; 29461; 3015.6. 

3. 32678; V337; ^/Msm2; 4^| ; (.098x85)*. 



308 ELEMENTS OF ALGEBRA. 

129. To Find a Number when its Logarithm is Given. 

I. If the Given Mantissa is Found in the Table. The first 
two figures of the required number will be seen on the same line 
with the mantissa and in the column headed N, the third figure will 
be seen at the head of the column in which the mantissa is found. 
Finally insert the decimal point as the characteristic directs. Thus, 

1. Find the number whose logarithm is 1.9232. 

Look for 9232 in tbe table. It is found on the line with 83 and 
in the column headed 8. Therefore, write 838 and insert the deci- 
mal point. Hence, the number required is .838. 

II. If the Given Mantissa cannot be Found in the Table. 

Find the next less mantissa, and the corresponding number ; also find 
the tabular diff'erence. Annex the quotient of the difference between 
the given mantissa and the next less mantissa divided by the tabular 
difference, to the corresponding number ; then proceed as before. 
Thus, 

2. Find the number whose logarithm is 2.7439. 

The next less mantissa is 7435, corresponding to 554. 

The next greater mantissa is 7443, corresponding to 555. 

.-. the tabular difference = 8. 

The diflference between the given mantissa and the next less man- 
tissa is 4. Since the given mantissa lies between 7435 and 7443, the 
corresponding number must lie between 554 and 555. An increase of 
8 in the mantissa causes an increase of 1 in 554. Therefore, an in- 
crease of 4 in the mantissa will produce an increase of ^, or .5, in 554. 
Hence, the mantissa 7439 must correspond to the number 554+ .5, or 
554.5. Therefore (II, Art. 121), write 05545 and prefix the decimal 
point. Hence, the number required is .05545. 

3. Find the number whose logarithm is 3.1658. 

The next less mantissa is 1644, corresponding to 146. 

The next greater mantissa is 1673, corresponding to 147. 

.•. the tabular difference = 29. 

The difference between the given mantissa and the next less man- 
tissa is 14. Annex \^, or .48 nearly, to the number 146, and insert 
the decimal point as the characteristic directs. Hence, the number 
required is 1464,8. 



LOGARITHMS. 309 

Exercise 112. 
Find the numbers whose logarithms are : 

1. 3.4683; 2.4609; 4.8055; 0.4984; 0.1959. 

2. 3.6580; 2.4906; 4.5203; 2.5228; 0.6595. 

3. 0.8800; 1.7038; 5.8017; 3.1144; 5.7319. 

130. An Exponential Equation is one in which the expo- 
nent is the unknown number ; as, iif = n, ifrf = n. Such 
equations usually require logarithms for their solutions. 

Example 1. Solve the equation 2F = 1.5. 

Process. Take the logarithm of each member, x log 21 = log 1 .6. 

By means of the table, 1.3222 x = .1761. 

.1761 
Therefore, x = . ^^aa = .1332, nearly. 

Example 2. Find the value of 3.208 X .0362 X .15734. 

Process, log (3.208 X .0362 X .15734) = log 3.208 + log .0362 
+ log .15734. 

log 3.208 =0.5062 
log .0362 =2.5587 
log .16734 = 1.1969 

2.2618 = log .01827. 
Therefore, 3.208 X .0362 X .15734 = .01827. 

Example 3. Find the fifth root of .05678. 
Process, log .05678 = 2.7542. 
5)2.7542 = 5)3.7542 + 5 

.7508 + T = T.7508 = log .6634, nearly. 



Therefore, -y/.05678 = .5634, nearly. 



310 ELEMENTS OF ALGEBRA. 

Example 4. Find the value of log^^ 144. 

Solution. To find loggi/g 144, is the same as solving (Art. 119) 
(21/3)' = 144, for I, squaring each side, etc., I = 4. 
Therefore, logg ^3 144 = 4. 

Exercise 113. 

Find by logarithms the values of the following : 

1. 360 X. 0827; 117.57 X .0404 ; i^ ; (31.89)3 



2. ^951; 380.57 X .000967; ^(•"^^^);^.y-Q"°^^^^ 

212.6 X 30.2 7435 -^343 
84.3 X 3.62 X .05632' 38731 X .3962' ^f2^' 



4. — ■ • ^ ; 72132 X .038209 ; -7.000313. 

^385.67 

5. (61173)*; -^; ^; ^X^.00l; Ip. 
^ ^ ' (.19268)i v'27 5^49 

Solve the followiug equations : 

6. 20" = 100 ; 2" = 769 ; 10" = 4.4 ; {^Y = 17.4. 

7. 10^ = 2.45 ; 5'-'" = 2"+^ ; a/S^^^ = '^/W^. 

8. 2* X 6*-2 = 52* X 7'-" ; 3^-' = 5 ; 4" = 64. 

9. (1)" =10; rrf = n\ m"*+* = 71 ; ?/i""' X c^"' = n. 

10. 2^+^ = 6^ 3^ = 3x2^+^ 31-0^-1^ = 4-1.^ 2^*-^ = 3^^-''. 

11. a2*^3y = ^5 ^3x^2. :^ ^^0. ^x,^5y = (^7)4 ^^^ = (^y)8 



LOGARITHMS. 311 

Find the number of digits in the values of: 

12. 312x28; 2^4; W^o . (4375)8. (396000)io. 

Find the number of ciphers between the decimal point 
and the first significant figure in the values of: 

13. (.2)*; (.5y«>; (.05)5; (.0336)io ; "x/sm, 

14. Given log x = 2.30103, find log xi 

Find the values of : 

15. loga 4 ; loga 8 ; log^ 32 ; log^ 128 ; log, 1024. 

16. log2 J ; log2 J ; logj ^ ; log^ ^^ ; log^ \^16. 

17. logs 729; log5l25; log, 625 ; log, 15625; log, J. 

18. log_e 1296 ; log_. - ^{^ ; log. ^{-^ ; logg^ 512. 

19. logs ^5- 125; log848 49; log8l28; loga^/s tJi- 

20. log,|/a^^^; log27^\; logj4; log, a ; log„|. 

21. If 8 is tlie base, of what number is § the logarithm? 
Of what J ? Of what 1 ? Of wliat 2 ? Of what 3 ? Of 
what If ? Of what 2 J ? Of what 3 J ? Of what Y ^^ ? 

22. In the systems whose bases are 10, 3, and J, of what 
numbers is — 5 the logarithm ? 

Find the bases of the systems in which : 

23. log 81 = 4; log 81 = - 4 ; log j^^ = 4; 
log iiftW = -4; log i^ = ± 2; log 1024- = ±571. 



312 ELEMENTS OF ALGEBRA. 



CHAPTER XXI. 



QUADRATIC EQUATIONS, 

131. A Quadratic Equation is an equation in which the 
square is the highest power of the unknown number. 

A. Pure Quadratic Equation is an equation which contains 
only the square of the unknown number; as, 5 x^ = 17. 

An Affected Quadratic Equation is an equation which 
contains both the square and the first power of the un- 
known number ; as, 5 ^^ — 2 a^ = 10. 

o , ^"^ + 5 ^ a; 17 
Example. Solve — I ^ = o + "t- • 

Process. Clearing of fractions, 12 a;^ + 60 — 9 a;^ = 4 a:^ -f 51. 
Transposing and uniting, x- = 9. 

Therefore, extracting the square root,* x = ± S. Hence, 

To Solve a Pure Quadratic Equation. Find the value of the 

square of the unknown number by the method for solving a simple 
equation, and then extract the square root of both members. 

Note. * In extracting the square root of both members of the equation 
a;2 z= 9, we ought to prefix the double sign (±) to the square root of each mem- 
ber; but there are no new results by it, and it is sufficient to w)1te the double 
sign before one member only. Thus, if we write ±, x = ± S, we have + x 
= 4-3, -^ X = — S, — aj = -j-3, and — x = — 3; but the last two become 
identical to the first two on changing the signs of both members. So that in 
either case, x = 3, and a? rr — 3. 



QUADRATIC EQUATIONS. 313 

Exercise 114. 
Solve the foUowiug equations : 

o 

3. a;(a;-10) = (6|-a:)10; {6 x + ^f = 756^ + 5x. 

35 - 2 a; 5x^ + 7 __ 17_-Jj; 
9 "*"5a,-2-7~ 3 

7. I (2 a; - 5)2 = 94 - 24x; 3 ar^ - 4 = ^±j? . 

8. -o = -o > rt ar + 7yj it' = a c^ + ??i x. 

3r — n or — m 

9. 2:2 ^ ^^aj_^ = ^j2:(l — wa;); 2 + 4.i'2 _. ^^(1 __^ 



10. 7^ — n X ■\- m = n X (x — \)', x VB + x^ = 1 + x^. 

-- mn — ic ?i — aa; . .-^ 5 

11. = ; X + vx^ — .3 = 



n — mx an — x ^x^ — 3 

12 ^ I ^ ^,. 1 , 1 ^^^ 

• 5+^ 5_a; > 1_>/1_^ l+Vl-a;2 a:2 



314 ELEMENTS OF ALGEBRA. 

132. Example 1. Solve x^ — 2ax + 4ab=: 2bx. 

Solution. Transposing, we have x^ — 2ax + 4ab — 2bx= 0. 
Arrange in binomial terms and factor, and we have (x — 2 a) (x — 26) 
= 0. A product cannot be zero unless one of the factors is zero. 
Hence, the equation is satisfied if x — 2 a = 0, or x — 2 6 = ; that 
is, a x = 2ay or if X = 2 b. 

Example 2. Solve ^ x2 + f x -f- 20^ = 42f + x. 

Process. Clear of fractions, transpose, and unite, 
3 x2 - 2 X - 133 = 0. 
Factor, (3 x + 19) (x - 7) = 0. 

Therefore, 3 x + 19 = 0, and x - 7 = 0. x = - 6^, and x =: 7. 
Hence, 

To Solve a Quadratic Equation by Factoring. Simplify the 
equation, with all its terms in the first member ; then place the fac- 
tors of the first member separately equal to zero, and solve the simple 
equations thus formed. 

Exercise 115. 

Solve the following equations : 

1. rz;2_i0;i;=:24; a?2+2a; = 80; ^2_ iga? + 32 = 0. 

2. a;2 + 10 = 13 (x + 6) ; a^ + 4x - 50 = 2 - 5 X, 

3. 4x2+13^+3 = 0; Zx^ -^ 1 = -\\x - x^ ^ ^. 

4. a;2-ic= 11342; 5a^+3x-4==8aj-7rz;2_2. 

5. \-Zx-x^ = 2x--^x-Z\ x^-2ax-\-%x = \^a. 

6. :i;3_5^2^5^ + 7^2. a^_|^ + _9_^0. 
x-\-Z 2x-Z 3-x 



7. lla:2_iii =9^. 



x+ 2 x-1 x-^2 



QUADRATIC EQUATIONS. 315 

8. (a;-2)(a;H9a: + 20) = 0; 2x^ + Sa^^2z-S = 0. 

3ar— 4 3a — 2a; 4 



10. mqx^ — mnx-\-pqx — n2y = 0; x+5 = Vx+ 5-\-6. 



11. {a-b)x^-(a + b)x + 2b = 0.; x^=21 + Vx^-^' 

133. An aflfected quadratic equation can always be solved by the 
method of completing the square. This method consists in adding to 
both memlxjrs such an expression as will make the memlier, with all 
the terms containing the unknown number, a perfect square. The 
explanation of this methocl depends upon the principle that a trino- 
mial is a perfect st^uare when one of its terms is plus or minus twice 
the product of the scjuare roots of the other two. This process 
enables us to extract the square root of the member containing the 
unknown number, and thus form two simple equations which may 
be solved separately. 

2x- 11 
Example. Solve ^(S-x) - ^^ = i (a? - 2). 

Process. Clear of fractions, transpose, and unite, 

-4x^ + 26x= 12. 
Divide by -4, ar»-J^a: = -3. 

Add * (l^)« to lioth members, x^^^x + (J^f = - 3 + (J^y = ^. 
Extract the square root, x — ^z=±^. 

Therefore, x-^ = ^, &nd x-^ = -^. x = 6, and x = f 

Every affected quadratic equation may be reduced to the 
general form 

7Ji2^ + 7ix -\- a = 0; 

where m, n, and a represent any numbers whatever, positive or 
negative, integral or fractional. Dividing both members by m for 

a n 

convenience, representing - by 6 and - by c, and transposmg, we 

have 




316 ELEMENTS OF ALGEBRA. 

x^ -\- c X = — b. 
Add (2) to both members, x^ + cx+i^j =-^+(2)- 

Or, x^+cx+ (0 =|(c2-4i). 

c 

Extract the square root, x + ^ = ±^ a/c'^ — 4 6. 

Therefore, x + ^ = ^ ^.c^ -4 b, and x + ^ = -^ ^c^- 4 6. From 



which a: = - 9 + | V^'"^ " 'I ^' ^"^ ^ = ~ 2 ~ ^ V^^ ~ ^ ^- "^^^^^ 



-c-l-Vc2-46 
values may be written in the form x — 5 • Hence, 



Common Method of Solving Quadratics. Reduce the equa- 
tion to the form x^ + c x = — b. Comflete the square of 
the first member by adding to each member of the equation 
the square of half the coefficient of x. Extract the square 
root of both member's, and solve the resulting simple 
equations. 



Notes : 1. * Always indicate the square of the expression to be added, in the 
first member. 

2. Since the squared terms of the square of a binomial are ahvays positive, 
the coefficient of x"^ must be made + 1, if necessary, before completing the 
square. This may be done by multiplying or dividing both members by — 1. 

3. The foregoing method is called the Italian Method, having been used 
by Italian mathematicians, who first introduced a knowledge of algebra into 
Europe. 



134. It is often convenient to complete the square without first 
reducing the simplified equation to the form in which the coefficient 
of x^ is 1. Thus, 

3ar-7 4a;- 10 7 
Example 1. Solve — - — + ^ _^ ^ = ^ • 



Process. 


1 = 7. 




3X7-7 

= + 


4 X 7 - 1(1 

7 + 5 " 


--h 




2 + i-- 


= i. 




J = 


--i- 



QUADRATIC EQUATIONS. 317 

Process. Clear of fractions, 

6 a:2 - 1 4 a: + 30 X - 70 + 8 a:2 - 20 ar = 7 a;2 + 35 a:. 
Transpose and unite, 7 ar'^ — 39 a; = 70. 

Multiply by 7X4, 196 ar^ - 1092 x = 1960. 

Add (39)2, 196 x2 - 1092 x + (39)« = 1960 + 1521 = 3481. 

Extiaet the si^uare root, 14 a: — 39 = ± 69. 

Therefore, 14a: = 39 + 59, and 14a: = 39-59. x = 7, and x = - Y- 
Verify by putting these numbers for x in the original equation. 

3 X - V - 7 4 X - V - 10 _ 
-1^ + -Y + 5 ~*' 

U - ¥ = I. 
i = i- 

When a quadratic e({uation appears in the general form 

»nx2 + nx + a = 0, 

the terms containing x may be made a complete square, without first 
dividing the equation by the coefficient of z*. Thus, 

Transpose a, mx^-{-nx = — a 

Multiply the equation by 4 m and add the square of n, 

4m^x^ -\- 4 7nnx -\- n^ = n^ — 4a m. 

Extract the square root, 2 wi x 4- n = ± ^/n^ — 4am. 

Transpose n, 2 w x = — »» ± >^n^ ~ 4am 

„, . — n ± \/w* — 4am 
Therefore, x = . Hence, 

Hindoo Method of Solving Quadratics. Reduce the equa- 
tion to the form in x^ + n x = — a. Multiply it hy f(/icr 
times the eoefficient of x^, and complete tlu sqtiare hy adding 
to each member the square of the coefficient of x in the given 
equation. Extract the square root of both member s^ and 
solve the resulting simple equations. 



318 ELEMENTS OF ALGEBRA. 

If the coefficient of x in tlie given equation is an even number, the 
square may be completed as follows : 

Multiply the equation by the coefficient of x^, and add to each mem- 
ber the square of half the coefficient ofx in the given equation. 

8x 20 

Example 2. Solve —7—0 "" o~ == ^* 
3. "t" z o X 

Process. Free from fractions, 

(3 x) (8 x)-20(x + 2) = 6 (3 x) (x + 2). 
Simplify, 6x^-5Qx = 40. 

Multiply by 6, 36 x^ - 336 x = 240. 

Add (Af )2, 36 x^ - 336 X + (28)'^ = 1024. 

Extract the square root,* 6 a; - 28 = ± 32. 

Transpose, 6 a: = 28 + 32, or 28 - 32. 

Therefore, x = 10, or - f 

Verify by substituting 10 for x in the original equation. 

8 X 10 20 

Process. _______ ^ 6, 

¥ -1 = 6, 

6 = 6. 
Verify by substit-uting — f for a: in the original equation. 

8 X - f 20 

Process. _^--^ _____ = 6, 

-¥ 20 _ 

- 4 + 10 = 6, 
6 = 6. 
Notes : 1. * We ought to write the double sign before the root of both 
members. Thus, ± (6 x - 28) = ± 32, tlie reason for not doing so is the same as 
given in Art. 131, Note. 

2. The Hindoo, or Indian Method, is supposed to have been discovered by 
Aryabhalta, a celebrated Hindoo mathematician, and one of the first inventors 
of algebra. It is not only more general in form, but much better adapted to 
the solution of equations in which the coefficient of the square of the unknown 
number is not 1. 

3. This method has an advantage over the common method in avoiding 
fractions in completing the square, and is often preferred in solving literal 
equa,tions. 



QUADRATIC EQUATIONS. 319 

135. In case the coefficient of the square of the unknown, in 
the simplified etjuation, is a square number the square may be 
completed as follows : 

Example 1. Solve 72 a: - 54 = (20 - z)(4x + 3). 
Process. Simplify, 4 x^ — 5 x = 114. 

""^•^ (i:Jb)' " 6)' ^ -' - ^ - + («' = Mj»- 

Extract the root, 2x- i = ±^. 

Transpose, 2 x = ^ + i,a. or | — ^. 

Therefore, z = 6, or — 4|. 

The coefficient of x^ may always be made a square 
number by multiplication or division. Hence, 

General Method of Solving Quadratics. Add to each 
member the square of the quotient obtained from dividing 
the second term by twice the square root of the first term. 
Tfien proceed as before. 



^ , 5 3 35 

Example 2. Solve —-r-, + - 



X + 4 X X — 2 

Process. Free from fractions, 

5 (x - 2) X + 3 (x + 4) (x - 2) = 35 (x + 4) x. 
Simplify, - 27 x^ - 1 44 x = 24. 

Divide by - 3, 9 x« + 48 x = - 8. 

(48 X \* 
— -= ) , or (8)«, 9 x2 + 48 X + (8)2 = 56. 

Extract the root, 3 x + 8 = ± 2 y/\i. 
Transpose, 3x = -8±2y'14. .♦. x= r-^^^ 



Note. The Common and Hindoo Methods of completing the square are 
modifications of the General Method. 



320 ELEMENTS OF ALGEBRA. 

Exercise 116. 



Solve the following 



1. 23a?= 120 + a:2; 42 + ^2^ 13 a;; 12 ic^ + ^ ^ 1. 

2. 22a:+ 23-2;2 = 0; a:2- |rz^=i 32; 2^2 + 3 2; = 4. 

3. a;+22-6 2;2=. 0; 25 ^=62:2+ 21; x2-2x = X 

4. 3x'^+12l = Ux; -^i-x=--^-x^; 91 a:2- 2a; = 45. 

5. 21 ^2 _|. 22 ^ + 5 = ; 9 ^2 - 143 - G a; = 0. 

6. 18a;2-27ir- 26 = 0; 50 a;2 - 15 a; = 27. 

7. 192;= 15-82;2; ^2+_4^^^1. ^2_l^_l3zzO. 

8. 5 2;2 + 14 2; - 55 ; (2; + 1) (2 2; + 3) = 4 2:2 - 22. 

9. 2(^-^)-3(2.' + 2)(.2;-3); .32^2 + 2.1 2: + | = 0. 
10. 252:+22;2= 42; |(2; + 6)(2;-2) - f (62jV + -V-^)- 

1 __ _1 ^ ^+16 11 _ 42: -171 

^^' IT^ 3^^~35' "T ■^¥~ 3 

42; x — 6_4:X+7 '^Jl^ 2! — 2 _ ^ 

^^- "9" "^ ^+~3 ~ ~T9~"' ^~^^ '^ ^^^S ~ ^^^ 



.0 _J__4^__L_. _i L-^1- 

3-x 5 9-2 2;' 2^-3 2:+5 18 



14. 



15. 



QUADRATIC EQUATIONS. 321 

4 3 4 5 3 



X — 2 X X -\- ^^ X — \ x-f2 X 

>y2+ 3 _ 12 + 5a^ 'dx 2a;-5 _^,^ 
-^^ "^aj2- 5 ~ 5(0:^-5)' 2:+ 1"^ 3a:-l~*^*8- 



^^ 5a;-7 a: - 5 3 a: - 1 , 
10. = = = -^ zTT^ ; -^ -—=, = 1 — 



Ix-b 2a;-13' 4a;+7~ x -\- 1 



,^ 12a:8-lla;'-*+ 10 a; - 78 ,, , 

17. 5-0 — ;, — -T-^ = l\x — h 



3a:+5 3x-5 _ 135 7 21 22 

3a;-5 3a; + 5~176^ x^+'^x^ '6x^-^x~ x 

\ 18 7 8 a; ic + 3 



19. 



«— 1 a; + 5 a;H-l x — b' x + % 2a;+l 



136. Literal Quadratic Equations. 

Example 1. Solve mx"^ -\-nx = — ; — -^ m x - n x^. 
Procesa. Transpose and factor, (m 4- n)x^— {m — n)x = — — • 

Multiply the equation by 4(m4- n) and add the square of (m — n)^ 

4 (m + nyx^ - 4 (m2 - n^) a: + (m - n)« = (m + n)«. 

Extract the square root, 2(m-}-n)a:— (m — n) = ± (to + n). 

Transpose, 2 (to + n) a: = to — n ± (m + »») 

= 2 TO, or — 2 n. 

«« * TO n 

Therefore, x - , , or 



TO + n' TO + n 



« « , 2a:+l 1/1 2\ 3x4- 1 

Example 2. Solve — r ( r — ~ ) = — :: — 

x\o aj a 



21 



322 ELEMENTS OF ALGEBRA. 

Process. Free from fractions, 

ax(2x+l)- (a-2b) = bx(Sx + l). 

Simplify and transpose, 

2ax^-i- ax -3bx^~bx = a-~2b. 

Express the first member in two terms, 

(2 a - 3b) x^ + (a - b) X = a - 2b. 

Multiply by 4 (2 a - 3 b), 

4 (2a-36)2a;2+4 (2a-36) (a-b)x = 8 a^-28ab + 24 b\ 

Complete the square, 
4(2a-36)2a;2+4(2a-36)(a-6)2: + (a-&)2 = 9a2-30a6 + 25 62. 

Extract the square root, 

2{2a - Zb) X + {a - b) ~ ± {3 a - bb). 
Transpose, 2 (2 a - 36) x = - (a - 6) ± (3 a - 5 6) 

= 2a-4&, or-2(2a-36. 

a-2b 
Therefore, x = ^^-^g^ , or - L 

1111 



Examples. Solve ^ + ^"i;-^ =-+ ^ ^ ^ 

111 1 

Process. Transpose, - - " = ^-+6 " j+^ 

Reduce each member to a common denominator, 

a — X X — a 



ax ia + b)(b + x) 
Free from fractions, {a-x){a-\-b){b + x)= ax(x- a). 

Transpose and factor, 

(a - X) [(a + b)(b + x) + ax'] = 0. 
Hence (Art. 11), a - a; = 0. .',x = a. 

Also, (a-\-b)(b + x) + ax = 0. 

Simplify and factor, 

^ ^ b(a + b) 

b(a + b-) + (2a + b)x = 0. .'. x= - ^^^^ ■ 

Kote. Always express the first member of the simplified equation in two 
terms, the first term involving x'^, the second involving x. 



QUADRATIC EQUATIONS. 323 

Exercise 117. 
Solve the following equations : 

-. 9 / . r\ . 7 f\ 2 . ^^^ m ax 

1. x^ — (a-{-b)x + ab = 0: mx^-] m = — 7 — . 

^ X a X b ^ mbx am z 

2. - + - = - + -; ma^ m = r 

a X b X a 

3. (a--6)r^-(a + 6)a;+26 = 0; a^ 3^ ^ abx=^ 21?. 

7^ X ^2a^ 2 z(a — x) _ a a^ + m2_ 
a^'^ b^~W' %a-2x ""V ^ ~ 

f>, ^ — n X •\- 'p X ^ n'p ■=■ ^ \ a^ -\- 2x Vn = n. 

6. a^3^ ~-2a^x+ n* -1 = 0; a^ -\- x (a - b) = ab. 

7. x^-\-mx-^cx-\-Ji^x+m^x = 0. 

8. ^a^x = (a^-b^ + xf; a^ {x - a)^ = 1)* (x + a)^ 

\a X J \ X a J 

10- —i . rxQ = (« — ^)^ ; ^— ^ X — X = —CX — Z. 

{a-\r by ^ ' b n 

11. («'~^(^J+l) ^2a?; 9a*6*2^»~6a8 63^ = 62. 



19 ^^+ <^ _ ^4-6 11 1 _^ 

« — 6~~na; — a' a a + a; a + 2a7~~ 



324 ELEMENTS OF ALGEBRA. 

a-x X b (a-l)^s^ -{- 2(Sa^l )x _^ 

lo. -r ■ " — — - ; ;; z — 1. 

X a — X c 4a — 1 

14. ra; + ^y = 4a:2; (ax-'^=\a?x\ 



137. Solution by a Formula. From the quadratic equa- 
tion moi?' -{■ nx — ^ a, 



— n± Viv^ — 4:am , ^ . 

*= 2Vv (^> 

By means of this formula the values of x, in an equation 
of the general form, may be written at once. Thus, 

Example L Solve 10ic2_ 23 a: = - 12. 
Process. Here, m = 10, n = — 23, and a = 12. 



c V .-. . .u 1 • /ix -(-23)± V(-2 3)-^- 4X 12X 10 
Substitute these values m (1), x = ^ y 10 

23 ±7 
~ 20 

= i or f 

^ r. -, 1 111 

Example 2. Solve r- ; — = j- + - h 

b + c -\- y c y 

Process. Free from fractions, transpose, and factor, 

(h + c)y^+ (h + cyy = -hc{h-[- c). 
Divide by 6 + c, y^ -^ (h+c)y = — he. 
Here, m = 1 , n = 6 + c, and a = hc. 



Substitute these values m (1), x = ~ 

-(b + c)±(b-c) 
~ 2 

= — r, or — 6. 

Note. In substituting the student must pay particular attention to the 
signs of the coefficients. 



QUADRATIC EQUATIONS. 325 

Miscellaneous Exercise 118. 

1. 17 a^ f 19 a* = 1848; Sa^ - 12x + 1 = 6x-2S. 

2. 5a:2 + 4^. = 273; ^x' + lx + ^ = 0, 

^•^+2^=& «(^' + 3)«=|(.+ 3)^-f 

4 x + 1 _ a;+2 _ 2a; + 16 x-2 
'^'^'^bx'^ 5 ~^' x-l~ x + 5 x + 1 

2a;+3 7 — x 7—3x 



5. 16x^-6x'-l = 0; 



2(2a:-l) 2(a:+l) 4-3a; 



a 5x_^ x-S 2 (a: + 8) ^ 3 a: + 10 
3"^'4~3a^a:-5"'" a:4-4 ~ a:+l 

7.--^=^; H(5^ + 36)2 = V^j(8a:2-4)^. 

X CL 

8. -4- = - + -; lla?»4-10aa: = ±a2; a:' + - + 2 = - 
p-\-x p x' 'XX 

,- ,2:2j2.fl,2fltJ 

9. ax ■\- dx^ — a-= d\ — « H = — o H 

ir^ c 7?r c 



10. WX2 



(w^ — w^ a; _ a^ a _ 



mn '3m — 2a 2 4a — (3 w 

11. a:^ - 2 a X = (6 - c + a) (6 — c — a). 

12. x^ - (a -{- b)x = \{7n -\- n -\- a -\- h){m -^ n-a-h). 

,o 1 _^ 1 a2 + a:2 ^^3 ^r - 3 

a -k- X a — X a* -- or x — 6 x -\- .* 



326 ELEMENTS OF ALGEBRA. 

14. ahx'^ - 2x{a + h) ^/'^ = {a -hf. 

' ^' "^ aW' ~ 18aH2 + 2ah 

a — 2h~x 5h — x 2 a — x — 19h_ 

a^ — 4:b^ ax + 2bx 2bx — a x ~~ ' 

^^1 1 m 2nx+ n 

^7 I . =: 

2x^-{-x~-l 2^2— 3 a; +1 2nx — n mx^ — m 



18. = ax. 

ax — Va^ x^ — \ ax + \o? x^ — 1 

Query. What is the diflference between the meaning of "the 
root of an equation" and "the root of a number"? 

138. Problems. The following problems lead to pure or affected 
quadratic equations of one unknown number. In solving such prol)- 
lems, the equations of conditions will have two solutions. Some- 
times both will fulfill the conditions of the problem ; but generally 
one only will be a solution. 



Exercise 119. 

1. Find a number whose square diminished by 119 is 
equal to 10 times the excess of the number over 8. 

Solution. Let x = the number. 
Then, x — 8 = the excess of the number over 8. 

Therefore, a:^ - 119 = 10 (a: - 8). 
The solution of which gives, a: = 13, or — 3. 
Only the positive value of x is admissible. Hence, the number 
is 13. 

Note. In solving problems involving quadratics, the student •should retain 
only those values for results that will satisfy the conditions of the problem. 



PROBLEMS. 327 

2. The difference of the squares of two consecutive 
numbers is 17. Find the numbers. 

3. Find two numbers whose sum is 9 times their differ- 
ence, and the difference of whose squares is 81. 

4. Find two numbers, such that their product is 126, 
and the quotient of the greater divided by the less is 3 J. 

5. Divide 14 into two parts, such that tlie sum of the 
quotients of the greater divided by the less and of the less 
by the greater may be 2r^. 

6. Find two numbers whose product is m, and the 
quotient of the greater divided by the less is n. 

7. Find a number which when increased by n is equal 
to m times the reciprocal of the number. Find the num- 
ber when n = 17 and vi = 60. 

8. Divide m into two parts, so that the sum of the two 
fractions formed by dividing each part by the other may 
be 71. Solve when m = 35 and n = 2^, 

9. Divide a into two parts, so that n times the greater 
divided by the less shall equal in times the less divided by 
the greater. Solve when a = 14, ?i = 9, and m = 16. 

10. A farmer bought some sheep for $72, and found 
that if he had received 6 more for the same money, he 
would have paid S 1 less for each. How many did he 
buy? 

11. If a train travelled 5 miles an hour faster, it would 
take one hour less to travel 210 miles. Find the rate 
travelled and number of hours required. 



328 ELEMENTS OF ALGEBRA. 

12. A man travels 108 miles, and finds that he could 
have made the journey in 4-|- hours less had he travelled 
2 miles an hour /aster. Find the rate he travelled. 

13. A number is composed of two digits, the first of 
which exceeds the second by unity, and the number itself 
falls short of the sum of the squares of its digits by 26. 
Find the number. 

14. A number consists of two digits, whose sum is 8 ; 
another number is obtained by reversing the digits. If 
the product of the two is 1855, find the numbers. 

15. A vessel can be filled by two pipes, running to- 
gether, in 22|- minutes ; the larger pipe can fill the vessel 
in 24 minutes less than the smaller one. Find the time 
taken by each. 

Solution. Let x = the number of minutes it takes the larger pipe. 
Then, x -\- M = the number of minutes it takes the smaller pipe. 

- =r the part filled by the larger pipe in one minute, 
and ^^ = the part fi lied by the smaller pipe in one minute. 

r^, P 1 1 1 

Therefore, - + 



X ' a; f 24 ~ 22f 
The solution of which gives, x = 36, or — 15. 
One pipe will fill it in 36 minutes, and the other in 1 hour. 

16. A vessel can be filled by two pipes, running to- 
gether, in m minutes ; the larger pipe can fill the vessel in 
n minutes less than the smaller one. Find the time taken 
by each. Solve when w = 56 and w = 66. 



PROBLEMS. 329 

17. B can do some work in 4 hours less time than A 
can do it, and together they can do it in 3| hours. How 
many hours will it take each alone to do it ? 

18. A boat's crew row 7 miles down a river and back 
in 1 hour and 45 minutes. If the current of the river is 
3 miles per hour, find the rate of rowing in still water. 

19. A boat's crew row a miles down a river and back. 
They can row m miles an liour in still water. It took n 
hours longer to row against the current than the time to 
row with it. Find the rate of the current. Solve when 
a = 5, w = 6, and n = 2. 

20. A uniform iron bar weighs m pounds. If it was a 
feet longer each foot would weigh n pounds less. Find 
the length and weight per foot. Solve when m = 36, 
a = 1, and n = |. 

21. A and B agree to do some work in a certain num- 
ber of days. A lost m days of the time and received n 
dollars. B lost a days and received c dollars. Had A lost 
a days and B m days, the amounts received would have 
been equal. Find the number of days agreed on and the 
daily wages of each. Solve when m = 4, 71 = 18.75, a = 7, 
and c = 12. 

22. A pei"son sold goods for vi dollars, and gained as 
much per cent as the goods cost him. Find the cost of 
the goods. Solve when m = 144. 

23. By selling goods for m dollars, I lose as much per 
cent as the goods cost me. Find the cost of the goods. 
Solve when m = 24. 



330 ELEMENTS OF ALGEBRA. 



CHAPTEE XXII. 
EQUATIONS WHICH MAY BE SOLVED AS QUADRATICS. 

139. In the equation m {if — yy + n (y^ — z/)^ -f a = 0, suppose 
(y^ — VY = 'C, then mx^ + nx -l-a = 0. Similarly, y^-Sy^ — 9 = 
may be changed to the form a:^ — 3 a: — 9 = 0. 

Hence, an equation is in the quadratic form when the unknown 
number is found in two terms affected with two exponents, one of 
which is twice the other ; as, a;^ + 5 a;^ — 8 = 0. 

The general form for an equation in the quadratic form is, 

ax^'' + b^f' + c = 0; 

where a, h, c, and n represent any numbers whatever, positive or 
negative, integral or fractional. 

Example I. Solve a;* - 13 x^ + 36 = 0. 
Process. Factor, (x + 2) (x - 2) (x + 3) (ar - 3) = 0. 
Hence, a; + 2 = 0, a;-2 = 0, a; + 3 = 0, and x — 3 = 0. 
Therefore, a; = ± 2, or ± 3. 

Example 2. Solve 8 a;~ ^ - 15 x~^ -2 = 0. 
Process. Factor, (x~^ — 2) (8 a:~^ + 1) = 0. 

^-i_2 = 0, ora:^ = i x = (hS^ = i ^2. 
Also, 8 X ^ + 1 = 0, or a;' = - 8. x= (- 8)^ = - 32. 

Example 3. Solve 3 a: + a:^ - 2 = 0. 

Process. Solve for x^. Thus, 

Multiply by 12 and transpose, 36 ar + 12 a:* = 24. 

Complete the square, 36 a; 4- 12 a:^ + 1 = 25. 

Extract the square root, 6 a:* + 1 = ± 5. 

Therefore, a;* = f , or - L 

Square each member, a: = ^, or 1. 



EQUATIONS SOLVED AS QUADRATICS. 331 

Example 4. Solve 2 -^^/^^ ^ 3 ^^ _ 55 _ q 

Process. Since -ij/x-^ is the same as x~ ', and /y/x-*^ is the same 
as x~ *, this equation is in the quadratic form. Transpose and mul- 
tiply by 12, 36 X-* + 24 x~^ = 672. 

Complete the sciuare, 36 z"* + 24 x~* + 4 = 676. 

Extract the square root, 6 a:~ * -f 2 = ± 26. 

x~^ = 4, or — ^. 

Therefore, a:' = J, or — ^. 

Extract the square root, a:* = db ^, or ± \/~ ^, 

Raise to the 5th power,* x = ± jij, or ± V(~A)*- 

Hotes : 1. When the roots cannot all be obtained by completing the square, 
the method by factoring should be used. Thus, in solving a;* + 7 a:* — 8 =: 0, 
by completing the square, we find but two values for x, re = 1, or — 2. Fac- 
toring the first member, we have (x + 2) (a* — 2 x + 4) (x - 1 ) {x^ + x-\-l) = 0. 
Hence, a; + 2 = 0, a^ _ 2 a: + 4 = 0, a; - 1 = 0, and x^ -f a: + 1 = 0. Solv- 



ing 



these equations, a: = — 2, 1 ± V^^, 1, and 



2. • In solving equations of the form a;» = a, first extract the wth root, 
and then raise to the »ith power. In practice this is the same as affecting the 



•quation by the exponent — . Thus, a: = a« . 



m 



Example 5. Solve a a:^" + 2» jr" = — c. 

Process. Multiply by 4 a, 

4 a* x^'* + 4a6ar"=: — 4ac. 
Complete the square, 

4a2j:2" + 4abx'* + b^ = -4ac+b^. 



Extract the square root, 2 a a* + 6 = ± yb^ — 4ac. 
Transpose b and divide by 2 a, af = — ^- — • 

Extract the -th root, x = [±VE^ILz*]" (i) 

Example 6. Solve Ax* - 37 x' + 9 = 0. 



332 ELEMENTS OF ALGEBRA. 

Process. Here, a = 4, 6 = — 37, c = 9, and n = 2. 

Substitute these values in(i), x = [ ^ V(-37)^-4X4X -9-(-37)f 
^^' L 2X4 J 

r ± 35 + 37 1^ 

= ± 3, or ± i 

Exercise 120. 

Solve the following equations : 

1. 0^4-14^2^-40; a;io + 312^5 = 32; x^-7x^ = S. 

2. 2^(19 + :i^3):^ 216; 2^2 + ^ ^.^ ,,2 + ^2 

3. 16(':c2+ i^ = 257; ^3 +14^^1107. 

4. 5 a:* + \/x = 22 ; 'v/^ + -|- = SJ. 

2 V 2- 

5. ^-t + 7 ^t = 44; 3 a;3 + 42 x^ = 3321. 

6. x^ + x^= 756 ; 3 V^^ _ 4 ^.^ = 7, 



^ . .. _2 . 2 

Vx ' xi 



7. 2V^ + ^^ = 5; 122:-t + f = 4 + ¥- 



a 3 ^t - a;-| + 2 = ; 2 a;-5 + 61 a^-t - 96 = 0. 

9. x-^ + ax-i = 2a^; x-^-2x-^ = S', x-^ + V^=(^ 

10. rr*" - |2;2» - || = ; 3 ict'* + 4 xl"" = 4. 

11. a:" + 13 o:''* = 14 ; 3 :r '"^ - 26 x ^'" = -16 



EQUATIONS SOLVED AS QUADRATICS. 333 

140. ?>iuatioijs may frequently be put in the quadratic form by 
grouping the terms containing the unknown number, so that the 
exponent of one group shall be twice the exponent of the other group, 
and then solved for the polynomial. Thus, 



Example 1. Solve a; - 3 x* - 4 Va: - 3 x* - 1 = - 2. 
The aquation may be put in the quadratic form if we reganl 
Vx — 3 a:* — 1 as the unknown number. Thus, 
Process. Add — 1 to each member, 



x-3a;*-l+4Vx-3ar*-l=-3. 



Put Vx -3x^-1 = y, y^ + 4y = -3. 

Therefore, y = 3, or 1. 

Hence, Vx - 3 x* - 1 = 3, or 1. 

Squaring, ' x - 3 x* — 1 = 9, or 1. 

Complete the square, x - 3 x* + } = ^, or Jf . 

Solving these equations for the values of x, we find x = 25, or 4, 

13±3v^ 
and X = ^ — — 

Hote 1. In solving equations of this fonn we must group the terms so that 
the expression outside of the radical, in the first member, is the same or a mul- 
tiple of the expression under the radical sign. 

Example 2. Solve x* - 6 ax« + 7 a" x^ 4- 6 a»x = 24 a*. 

Process. Add 2a«x^ x*-6ax^+9a^x^-^6a*x = 24 a* -f 2a«x«. 
Transpose 2 a«x2, x*- 6 a x«+ 9 a^ x2+ 6 a«x - 2 a^x^ = 24 a*. 
Group and factor the terms, 

(x2 - 3 ax)2 - 2 a2 (x2 - 3 ax) = 24 a*. 
Regard x*— 3 ax as the unknown number, and complete the square, 

(x2 - 3 a x)a - 2 a« (x2 - 3 a x) -f- a* = 25 a*. 
Extract the square root, (x'^ — 3 a x) — a^ = ± 5 a* 

Therefore, x« - 3 a x = 6 a", or ~ 4 a«. 

Complete the square and solve, x = 2 (3 ± \/33), 



334 ELEMENTS OF ALGEBRA. 

Note 2. Form a perfect square with xi and —6a x^. The third terra of the 
square is the square of the quotient obtained by dividing 6ax^ by twice the 
square root of x*. 

Example 3. Solve a:2 + 4x — 4a:r-i-fx-2 = ^. 
Process. Use positive exponents, rearrange terms, and factor, 
a=^ + ^, + 4(x-i)=|. 

Regard x as the unknown number, and subtract 2 from both 

sides, „ ^ 1 . f 1\ 11 

Factor, and complete the square, 



(.-iy+4(.-^)+4=^. 



Extract the square root, a: — - + 2 = ±f. 

Therefore, x — - = — ^, ot — ^. 



X 



Free from fractions, ^ ar^ — 1 = — ^ a:, or — ^ x. 

Complete the square and solve, x = ^ (— 1 ± y'S?) , 

x = ^(-ll±^/T57). 

Note 3. Form a perfect square with x^ for the first term and — for the 

third. The middle term will be twice the product of their square roots taken 
with a negative sign. 

A Biquadratic Equation is an equation of the fourth de- 
gree. Biquadratic means twice squared, and hence the 
fourth power. 

If a biquadratic is in the form, 

x^+2mx^+ (m2 + 2 n) a:^ + 2 mnx = a (ii) 

the first member becomes a perfect square by 

Adding n^, or the square of the quotient obtained by dividing the 
coefficient of x by the coefficient of x^. 



EQUATIONS SOLVED AS QUADRATICS. 335 

Thus, extracting the stiuare root of the fii-st member, 

X* i- 2mx^ + {inr +'2,n)x^ -\- 2mnx | x'^ + mx -{- n 

X* 

2x^-\- mx I 2mx^+ (m^ + 2n)x^-\-2mnx 

2mx*+{m^ )x^ 

2x^ + 2mx-\-n\ + ( 2n)x^ + 2mnx 

+ ( 2n)x^+ 2mnx + n^ 

— n*. Hence, 
the equation may be written, 

{x^ -\-mx + n)2 - n^ = a, or (z" + w z + «)« = a + n^ (iii) 

Example 4. Solve x* - 10 «« + 35 z^ _ 50 x = 1 1. 

Process. Here, 2m = — 10, 2 win = —50. .-. m = — 5 and n = 5. 

Since m^ + 2n = 35, the equation has the form of (ii). 

Add 25 ; or put w = — 5, n = 5, and a = 1 1 in (iii), 

(z2 - 5 z + 5)2 =r 36. 

Extract the square ^oot, z* — 5z + 5 = ±6. 

Therefore, z^ - 5 z = 1, or - 11. 

5 ± a/29 
Complete the square and solve, z = ^ — , 

5± a/^Hq 
^ = 2—- 

Kote 4. After adding the value for n* the first member may be factored by 
substituting the values for m and n in (iii). 



Exercise 121. 
Solve the following equations : 

1. (3^ + x-'2)^-lS(2^ + x-2) + Z6 = 0. 



2. a^» 4- 2 a; + 6 Va^+2x+5 = 11. 

3. 2:2 + 24 = 12 V^-qrf. 2a: + 17 = 9 V2 a; - 1. 

4. a:2 - a: + 5 (2a:2 - 5 a; + 6)i = J (3 a: + 33), 



336 ELEMENTS OF ALGEBRA. 

^ (a;2 _^ ^,. ^ 6)i ^ 20 - |(a;2 + a; + 6)^ 

7. (« + l)%4(«+!) = I2. 

8. (a;2 - 5 a;)2 - 8 (ic2 _ 5 ^) ^ 43^ 

9. 9 ^ - 3 ^2 _!_ 4 (^2 _ 3 ^ + 5)^ ^ il^ 

■«(-9"-!(-^)-S- 

11. (3a:2-10^+ 5)2-8(3^- 10a^+ 5) = 9. 

13. :i:4 + 6^^ + 5:z;2- 12a;=12. 

14. x^-Qa^-2^x^+ 114 2; = 80. 

15. ^4 + 2 2^ - 25 a;2 _ 26 a; + 120 = 0. 

16. 2:4-8 .>;3 + 10 2:2 + 24 a: + 5 = 0. 

17. a:4 + 8 2:3 + 2 a:2 - I a: = |_ 

18. (^3 _ 16)1 _ 3 (2^3 - 16)i = 4. 

19. ^-Y^^-Vx-x-^^A.; 2;2+3:^-32:-i + 2r2 = ^. 



EQUATIONS SOLVED AS QUADRATICS. 337 

141. Equations Containing Radicals may be Solved. Thus, 

Example 1. Solve x - ^3* + 2 x + 12 + 2 = 0. 

Process. Transpose, x + 2- ^x» + 2x -f- 12. 

Raise both members to the third power, 

a:» + 6 x-» -f 12 X + 8 = a:* + 2 X -h 12. 
Transpose and simplify, 3 a:"^ + 5 x — 2 = 0. 

Factor and solve, x = ^, or — 2. 

Verify by putting these numbers for x in the original equation. 

Process, x = J. x = 

2 - ^-8-4+12 + 2 = 0, 

-2-0 + 2 = 0, 

= 0, 

1 1 



i - ^tjV + f + 1^ + 2 = 0, 

i - i + 2 = 0, 

= 0. 



Example 2. Solve 



Vx* + 1 ^/x'^ - 1 ^/J^ - 1 



Process. Multiply by /^/x* — 1, 



Vx2 - I + yx« +1 = 1. 
Square, x^ - 1 + 2 ^/x'^~^\ + x« + 1 = 1. 

Transpose and simplify, 2 ^/x* — 1 = 1 — 2 x^. 

Square, 4a:*-4 = l-4x« + 4x*. 

Simplify, x« = f . 

Extract the square root, x = i J y'S. 



Exercise 122. 
Solve the following equations : 



1. 3 V^+6 + 2 = a;+VT+6; a:+ \/iT~2 = 10. 

2. aj 4- 16 - 7 v./- + 16 =10-4 Vx + 16. 

3. 2a:+ V4a; + 8 = J; V4a;+ 17+ V^l » 4. 



338 ELEMENTS OF ALGEBRA. 



4. 2'v/3a; + 7 = 9-V2i^-3 



- V4:X + 2 4 - Vi 



4A/ic Vx 



12 5x- 9 _ \/5^-3 

Vic+ 12' V5^+3 2 

44-2; 



5. V^ + ^ = y ; -7= — ;==! + 

Vic + 12 V a; + 3 

6. Vi^-2\/^-=2;; 1^64 -{-2x^-Sx- ,, 

V4: -\- X 

H, » A/ . * A/ o /o— 3.a; — 1 . , Vi^a; — 1 



+ ^ = a;. 



X + V'2 — x^ X — V2 — x^ 



^ V7 y2 + 4 + 2 a/3 7/ - 1 ^ m- Vrn^ - y^ 
a/7 2/^ + 4 - 2 A/3y- 1 ' ^^^ + A/m2 - 2/2 

10. ^a2 + 2a^2_2aaj = -^^iL; A/6a:-a;3= "^ 



11. 



Va + .^' A/a; 

a^-&2 V^ + & a/.^: + 9 3a/^-3.8 



a/^ 2; + & ^ ' Vx 9 — Vic 

^^ /— -- / 12a 6 + SVx 4 

12. Va + X -{■ ya — x= ; 7— r — 7=^ = "7= 

5 A/a + a; 4 + Y^,^ Va; 

13. V^f^+Jj - Vy^^ = V2y; 2rr+3A/^=27. 

, , a; + a/S 2:2-2; 12 + 8 a/S 

14 ^ — — — • X — -z 

X- Vx 4 ' x-b 

o_^ + ^'^^. a; _ A/a; — 12 
2; '4 2: — 18 



16. A/2;^ + A/2r^ = 6 a/5; Vx — a ■\- "sJx -^ a = ^/'^ 



THEORY OF QUADRATIC EQUATIONS. 339 



THEORY OF QUADRATIC EQUATIONS. 

142. Representing the roots of mx^ + nx = ~ a by r and r^, we 
have (Art. 134), 



— n + \/n^ — 4 a m ^ 




— n - \^n^ — 4am 




'1- 'Zm 




n 

' + '. = -» 


(i) 


a 

rr, — — 

*■ m 


(ii) 



Adding, 

Multiplying, 

Hence, if a quadrntic appears in the form, mx^ + nx = — a^ 

I. The sum of the roots is equal to the quotient ^ with its sign changed, 
obtained by dividing the coefficient ofx by the coefficient ofx^. 

II. The product of the roots is equal to the second member, with 
its sign changed, divided by the coefficient of x^. 

By means of (i) and (ii) the ori«,Mnal equation becomes, 

m x* — wi (r -f r,) a: 4- m r Tj = (1) 

Factor, m(x-r)(x - r,) = (2) 
If m = 1, x^-\-nx = — a 

x^- (r + rj) X + rrj = (iii) 

ix-r)(x-r,) = (3) 

If the roots of a quadratic equation be given, by means 
of (iii) we can readily form the equation. 

Example 1. Form the equation whose roots are ^, —\. 
Process. Here, r = ^ and r^ = -~ \. 

Substitute these values in (iii), x^- (^-\)x + (\) (-\) = 0. 
Simplify, 8 x« - 2 x - 1 = 0. 

Example 2. Find the sum and the product of the roots of 
8x«+3x-5 = 0. 

Process. Here, m = 8, n = 3, and a = — 6. 

Substitute in (i) and (ii), r + r j = — { and rrj = — j. 



340 ELEMENTS OF ALGEBRA. 

Exercise 123. 

Find the sum and product of the roots of : 

1. 2:2 + 8^ = 9; 12 a:'^ - 187 ^" + 588 = 0. 

2. 20x-^ = 5-5x-^l a^-6x+9 = 9x. 

3. 32:2 + 5 = 0; ^24.^2:=^^. a^-l5x = S. 

. „ 2mn^x mn 207, 2 . 1.2 n 

4. x^ = ; x^ — 2 X — 0? -\- h^^ = 0. 

m — n m — n 

Form the equations whose roots are : 

5. 7,-3; |,-|; 5, -3; ± V=^; 2- V3, 2 + V3. 

6. 0,-5; 7 + 2A/5, 7-2a/5; 1 + V2, 1 - V2. 

7. 7/1 (m + 1), 1 — //I ; — , ; 1- , — a. 

^ ^ n m a — h 

8. - w + 2 \/2 /I, - w - 2 V2 71, > -^— 

143. A Root is said to be a Surd when it can be found 
only approximately ; as, a; = =t ^^, 

Real Roots are values of the unknown numbers that can 
be found either exactly or approximately. 

Imaginary Roots are values of the unknown numbers 
that cannot be found exactly or approximately; as, 

x = ± V^^. 
Character of Roots. For brevity, represent the roots of the 
equation mx^-^nx + a- by r and r^, then, 

r= TJ}L , 

2m 

_ — n — \/7i^ —4 am 

^^ 2m "' 



THEORY OF QUADRATIC EQUATIONS. 341 



It is seen that the two roots have the same expression, y/n^—Aam. 

If n^ is greater than 4 am, n* — 4 a m will be positive^ and 
\/n* -4am can be found exactly or approximately. 

If n is positive, r^ is numerically greater than r ; if n is negative, 
r is numerically greater than Tj. Hence, 

I. Condition for Eeal and Different Ebots. n* - 4 a m, 

positive. 

nitiatration. 3a:2-2x + | = 0. 

Here, wi = 3, n = — 2, and a = |. 

n2-4am= (-2)2 -4X^X3=4-1 = {. 

Therefore, the roots are real and different. 

Evidently both roots will be rational or both surds according as 
n* — 4 a m is, or is not, a perfect square. Hence, 

11/ Condition for a Rational or a Surd Root, n^- 4am, 

a square number; or, /y/n- — 4a7/t, a surd. 

lUustrationB. (1) z* - 3 x - 4 = ; (2) 8 x^ + 5 x - J = 0. 

(1) Here, m = I, n = - 3, and a = — 4. 
n2 - 4 am = (- 3)* - 4 X - 4 X 1 = 9 -H 16 = 25. 

Therefore, the roots are real and rational, and dilferent. 

(2) Here, m = 8, « = 5, and a — — \. 

Vw* -4am = \/25 + 8 = \/33- 
Therefore, the roots are real and surds, and different. 



If n* is less than 4am, n'— 4am will be negative, and \/«*— 4am 
will represent the even root of a negative number. Hence, 

III. Condition for Imaginary Roots. n«-4am, negative, 

Uluatration. 2x«-3x + 2 = 0. 

Here, m — 2, n = — 3, and a = 2. 

n«-4am=(-3)« -4X2X2^9- 16 = -7. 
Therefore, the root.<< are both iniM^'inary. 

If n* = 4am, n*-4«m = 0, and the roots will be real and equals 
and have the same sign, but opposite to that of n. Hence, 



342 ELEMENTS OF ALGEBRA. 

IV. Condition for Equal Roots, n^ ~4am = 0. 

Illustration. 4x2 — 12a; + 9 = 0. 

Here, m = 4, n = — 12, and a = 9. 

71^ -4 am =144- 144 = 0. 

Therefore, the roots are real and equal. 

If a m is positive, for real roots, n^ — 4am will be positive and 
less than n^, since ^n^ — 4am will be less than n. 

If a 771 is negative, ^n^ — 4 am will be greater than w, since 
n^ — 4 a m will be greater than n^. Hence, 

V. Condition for Signs. // a m is positive, real roots have the 
same sign but opposite to that of n. If am. is negative, the roots 
have opposite signs. 

Illustrations. (1) 2x2-10a;+12 = 0; (2) 2x^-5x-3\^ = 0. 

(1) Here, m = 2, n = - 10, and a= 12. 

n2 - 4 a m = 100 - 96 = 4. 
Therefore, the roots are rational and positive, and different. 

(2) Here, m = 3, n = — 5, and a = — S^J. 

n2 -4 am = 25 + 47 = 72. 
The roots are surds and have opposite signs, and different. 



Exercise 124, 

Determine by inspection the character of the roots of : 

1. 5x'^-x = 3; 7x^+2x=-^: Ux^-x^-^, 

2. 4^2+ 52a; = 87; 3 x^ + 4a' + 4 = 0. 

3. 6-llx-9x'^ = 0; 9a = 3 + 4la;2 

4. 10x+S:^ = -3x^- lx'^-^x + l=0. 

5. 3a:2-2a:+3=:0; 4:X^-3x-5 = 0. 



THEORY OF QUADRATIC EQUATIONS. 343 

8. 6ar2+ 5 2:-21 = 0; 13 ar^ + 56 a; - 605 == 0. 

9. 9 ar^ - 30 a; + 41 = ; 40 o,^ - 100 a; - 360 = 0. 

Query. How many roots can a quadratic equation have ? Why ? 

Miscellaneous Exercise 125. 
Solve the following equations : 
1. a:t + 7J = 44; x-^-2x-^ = S; 3s^-:^-2 = 0, 



2. k/t-^- + J- = 2X- 1 + 8 a:* -h 9 '>y^ = 0. 

▼ i — X ^ X 

3. —== — 2 V2 a: = 59 ; ., ^ , ._ + — = = 3 Va;. 
^/2x 1 + 5 Va; Vx 

4. a;*"-2a:3« + ^ = 6. 3^-2o(^ + x = a. 

5. a^ + -^a:3_39^ = 81. a^ _ 2 a^ + a; = 380. 

6. 108a:*= 180^8- 20aj- Sla^H- 7. 

7. a:*-10a:2 + 35 a:^ _ 50^; = - 24. 

8. {x - rt)f + 2 v^ (a; ~ rt)i - 3 7i = 0. 

9. a:t ~ 4 a:f + r- J + 4 a:-f = - {. 



344 ELEMENTS OF ALGEBRA. 



V y -\- 2 a — ^y — 2 a _ y x -\- ^x _ o^ — x 
^y-2a + V2/ + 2a ~" 2 a' x - '^x 4 



4:r" 



11. 3a;"'^a;"-^:=:4; V'6:r+l + K2: + 4+ V^^+l = 2. 



12. a; V5 + a/2 a' 4- 2 = V^ + :^; a: - 1 = 2 + — - • 

13. 2 Vi + 2 :z^-i = 5 ; 6 Vi = 5 :c- 2 - 13. 

14. x^ + 2 m^x-^ = ^m; oT^ + 2 = ^ ,"^ • 

X- 3 + 5 

15. ^^2;+ V2^^=T-Kx- V2^T = I v/ — ^^^=. 



ic 



+ Va^ - 1 a: - V:r2 - 1 



16. ^ ^ ^ -^ - :: ^ = 8 ^' Va:^ - 3 ^^ + 2. 

a; — Vx^ — 1 2: + Va;2 — 1 



17. — ^^ — ^~^, ^^-— = & ; ic3 + a; A/a; - 72 = 0. 

ft + 2; 

18. State the conditions that will make the roots of 
x^ + Ax + B = 0: (i) surds; (ii) real; (iii) imaginary; 
(iv) equal; (v) have same signs; (vi) have opposite signs; 
(vii) equal in value but opposite in sign. 

19. Find a number such that if its nth root be increased 
by one half of its ^th root, the sum shall be a. Solve 
when n = 2 and a = 5. 

20. Find a number sucli that if its nth power be dimin- 

2 a . ' 

ished by the - th root of the th part of it, the remainder 
-' n c ^ 

shall be m. Solve when m = 144, n = 2, a = 27, and c = 5. 



SIMULTANEOUS QUADRATIC EQUATIONS. 345 



CHAPTER XXIII. 



SIMULTANEOUS QUADRATIC EQUATIONS. 

144. Only certain forms of quadratic equations involving two 
unknown numbers can be solved. Thus, 

Example 1. Solve the equations : ^ ^ ^„ ^~ « « ^. ^2 

10 - 1/ 
ProcesB. From (1), x= ^ (3) 

/l()_y\2 /10-w\ 

Substitute in (2), 2 ( — ^-^ j - ( "^ ) 2/ + 3 3^« = 54. 

Simplify and factor, (y — 4) (4y + 1) = 0. 

Therefore, y = 4, or — ^. 

Substitute in (3), ar = 3, or 5^. Hence, 

When one of the Equations is of the First Degree. Solve 
by substitutiou. 

The Degree of a term is the mimber of literal factors 
involved, and is always equal to the sum of their ex- 
ponents. 

Each literal factor is called a Dimension. 

Thus, 3 xy is of the second degree, and has two dimensions. 
5 x«j/* is of the fjih degree, and has Jive dimensions. 

« o 1 .1- *• 5l83xy + 72x+36y = 88 (1) 

Examples. Solve theequations : | ^^^^^^^^^^3^^ ^ g^ ^^^ 

80-36y .^x 
Process. From (2), a; - Y77— jTgo ^' ^ 

Substitute in (1), 

183(80y-36y«) 72(8<l-36y) , _„ _ ^^ 
177y + 60 + 177 y 4-60 + ^^ ^ " ^^^ 
Simplify, 9 y* + 57 y - 20 = 0. 

Complete the square and solve, y = J» or — 6f . 

Substitute in (3), ar = ^, or - f 



346 ELEMENTS OF ALGEBRA. 

Example 3. Solve the equations . | 6 x^ - x - 3 3/ = 5 (1) 

^ lx^ + x-y=l (2) 

Process. From (1), y = :: (3) 

Substitute in (2) and simplify, 

3a:2-4a:-2 = 0. _ 

Complete the square and solve, x = :z 

^ , . . , . 11 ±7 ViO ,, 
Substitute m (3), y — ^ . Hence, 

When each Equation Contains only one Second Degree 
Term, and that Term Consists of the Same Product or Square 
of the Unknown Numbers. Solve by substitution. 

Exercise 126. 

Solve the following equations : 

C^xy = 50. C2x + y = 22. 

2. |3.;j7/+6^-2z/ = 4. g {x-y = ^. 
\4.xy — X + ^y =^l. ' \xy=126. 

^ (x + xy = 24. ^Q r 2^3 -7/ = 218. 

' \xy + y = 21. ' \x~y = 2. 

4 f2y2 + y:=28. ^^ fx--y=:4. 

^ (15 + y = x. ^2 (x + 3y=16. 

\xy = 2ij^. ' \3x^-h2xy-y^ = -12 

Q^ (xy+Qx + 7y = 66. ^3 {frJ^ + y2='iS5. 
\Sxy + 2x+5v = 70. ' lx-v=3. 



SIMULTANEOUS QUADRATIC EQUATIONS. 347 
15/2^ + ^ = 9. .Q (x-y = 4. 

145. All equation containing two unknown numbers is 
symmetrical when the unknown numbers can change places 
without changing the equation ; q^q, 'i a? — 4: x y ■\- Z 1/ = 2 \ 
ic* 4- b 3^y -\- 5 xij'^ + y^ = — 5 x^ ?/. 



Example 1. Solve the equations : 


( x^ + ,/ = 89 (1) 
Uy = 40 (2) 


Process. Add (1) to twice (2), 
Siibtract twice (2) from (1), 
Extract the square root 0! (3), 
Extract the square root of (4), 


x2 + 2xi/-f 2/2^ 169 (3) 
x2-2xy + y2-9 (4) 

x + y = ± 13. 
X - 1/ = ± 3. 


We now have to solve the four pairs 


of simultaneous equations, 


ar-f t/=13> x + y= 13 > x -f y 
X- y= 3r x-y = -zy x-y 


r^--13) x+y = -137 
= 3 y x-y = - 3>' 



There are four pairs of values, two of which are given by x = Jb 8, 
'/ = ± 5, and the other two by x = ± 5, y = ± 8, in which the upper 
signs are to be taken together, and the lower signs are to be taken 



together. 



Kotes : 1 . If the second members of two simple equations have the sign ± , 
we will have six simultaneous .simple equations to consider. 

2. The above equations may be solved as in Art. 144, but the symmetrical 
nwtliod is more simple. 

K\.\Mi'LE 2. Solve the equations: \ „ ^~ „ _, \J. 

^ lx^-xy + y^-2l (2) 

Process. Divide (1) by (2), x + ?/ = 6 (3) 

Square (3), x^ + 2xy-\-y^ = 36 (4) 

Subtract (2) from (4), 3xy= lb, or xy = b (6) 

Subtract (5) from (2). x^ - 2 x.V + y2 = 16. 

Extract the square root, x — y = ± 4 (6) 

Add (3) and (6) and divide the result by 2, x = 5, or 1, 

Subtract (6) from (3) and divide the result by 2, y = 1, or 5. 



348 ELEMENTS OE ALGEBRA. 

Example 3. Solve the equations . 5 ^'^ + 2/^ - ^ - 2/ = "8 (1) 
^ ixy + x + y = 2^ (2) 

Process. Add (1) to twice (2), x'^+2xy + y'^+x + ij = 156. 
Factor, {x + yy + (x + y) — 156. 

Regard a; + ^ as the unknown number, complete the square, and 
solve, * a; + ?/ = 12, or- 13 (3) 

Subtract twice (2) from (1), factor, and transpose ^(x + y), 

{X - 3,)2 = 3 (X + y-) (4) 

From (3) and (4), {x - yY = 36, or - 39. 

Therefore, x - y = ± 6, ov ± y.1~39 (5) 

- 13 ± V-39 
Add (5) and (3), etc., a; = 9, or 3, and ^ 

Subtract (5) from (3), etc., y = 3, or 9, and -^ • 

Example 4. Solve the equations : < ^ ^ ^ 

^ lx + y = 4 (2) 

Process. Raise (2) to the fourth power, 

x*-\-4x^y + ex^y^+ 4xy^ + y^=256 (3) 

Subtract (1) from (3), etc., 

2x»y + 3x^y^+2xy^- 87 (4) 

Square (2) and multiply the result by 2 xy, 

2x»y-h4x^y^-^2xy» = 32xy (5) 

Subtract (4) from (5), etc., x^y^- 32xy = - 87. 

Regard xy a.^ the unknown number, complete the square, and 
solve, xy = 29, OT 3. 

We now have the two pairs of equations to solve, 
X -^y= 4\ X + y= 4) 

xy=29) ' _xy = 3} 

^ ar = 2 ± 5 \/^> 

From the first, A "^ ^ 

l2^^2T5V-l. 



(0; = 3, or 1. 

' \y=\, or 3. 



From the second, ^ 

When the Equations are Sjrmmetrical Combine them in 
such a manner as to remove the highest powers of x and y. 



SIMULTANEOUS QUADRATIC EQUATIONS. 349 

Exercise 127. 
Solve the followiug equations : 

' \x + 7j = n. ' \x^-xy + y^=2l. 

^ ix^ + x + i/+i/=lS. g (2^+x^y^+y^ = 9Sl. 
' \xy = 6. ' \x^ + X y -\- i/ = 49. 

^ (x^y^+2x+2y=50. ^Q (x^-xy + y^=U. 
\xy + x + y = 2d. ' \x + y=U. 

g (2^ + yi = 52. j^ ra^4-a^»v/2 + /=133. 

\ X + y + xy = 34. ' \ o^ + x y + y^ = 19. 



6. <^ 2:2 + 2r^ - 900 • 12. ^ a?» + 2r* ~ ^ * 

l«y = 30. U + y = 8. 

146. An algebraic expression is said to be honwgeneovs 
when all its terms are of the same degree. 

Thus, 9 x*<> -f 3 X y* — 8 r* y* is homogeneous, for each term is of 
the 10th degree and has ten dimensions. 

Example 1. Solve the equations H « « « „ « « \^i 

Process. Let y — vx, .iiifl substitute in both equations. 
From (I), 6x2-f- 2t'«z«-6t;x«= 12. 

12 
Therefore, ^'= 6-5»; + 2 »;« ^^^ 

From (2), 3x« + 2yx2 = 3 r^x^* - 3. 

Therefore, ^' = 3t;«-2t;-3 ^"^^ 

12 3 

Equate (3) and (4). e-5v + 2t^ = 3ra-2t;-3 

Simplify and solve for », r = }, or - 1. 



350 ELEMENTS OF ALGEBRA. 



Substitute y = | 



12 

= 4. 



6-5 X 1 + 2(1)' 



Substitute v = — ^ in (3), 

12 25 

"^ 6-5X-| + 2(-f)2-3l' 
.-. x = i /j V31. 
3/ = -f^-T^\V31. 



Notes : 1. In finding the last values of x and y, it will be observed that ± 
values of x gives respectively — and + values of y. This indicates that the 
equations can be satisfied only by making y ^ — ^^ V2>1, when x^-\- ^^^ V'6\ ; 
and when x =. — ^^ V31, y must be + ^f VZl. 

2. The sign T denotes precedence of the negative value. 

When each Equation is of the Second Degree and Homo- 
geneous. Substitute v x tor y in botli equations. 

Exercise 128. 

Solve the following equations : 

^ ^ x^ + xy = lb. n ( x^— o xy + y'^ = — 1. 

' \y^ + xy= 10. ' \Sx^-xy + Sy'^ = lS. 

2 (x''-xy = 24:. ^ (2x^-5xy+3y'^=l. 

\ X y — y^ — 8. * \ 3 x^—5 xy+'2y'^ = 4=. 

= 21. 
18. 



2 (x^ + 4:Xy=lS3. g (x^~2xy 

' \4:xy + 16/ = 228. * I xy + y^ = 

^ (2x^+3xy=26. ^ (x^+3xy = 

{Sy'^+2xy= 39. ' \xy + 4:y^ = 

^ ( 4^2_^to/+4?/2=13. . ^ ( x^ + xy + 2 y^ 

' I 8x'^-12xy+Sy^=ll. [2x^+2x1/ + y^ 



= 54. 
115. 



74. 
■12^?/+8/=ll. " 12^24.2^^ + ^=73. 

Queries. What is a homogeneous equation % Into what forms 
may simultaneous quadratic equations, M'hich can be solved, be 
grouped % What is the degree of the equation arising from eliminat- 
ing one unknown number from two equations, each of the second 
degree ? Prove it. How may such equations be solved .^ 



SIMULTANEOUS QUADRATIC EQUATIONS. 351 

Note. In solving the following equations the student is cautioned not to 
work at random, but to study the equations until lie sees how they may be 
combined in oixler to produce sinjple equations, and tlien perform the opera- 
tions thus suggested. Usually the operations of addition, subtraction, multi- 
plication, division, or factoring will effect a simplification of the equations. 



Miscellaneous Exercise 129. 
Solve the following equations : 



\xy= 15. 


11. 


^2^+:\u.^y-\-:\xy^-\-2y^ = {). 
U2-r.t// + y2^1-x'^/. 


2 /^-3' = 3- 


12. 


{.ly-\- .I25x = y — x. 
I y — .0 X = .7o X y — o X. 


^■\x-.>,= l. 


13. 


f .Sx + .l2by = ox-y. 
1 3 a: -\-y = —2.25xy. 


<x> + y' = 2U. 
^ \x + y = 22. 


14. 


1 a:* -f 7/4 = 706. 
\x + y = 2. 


(a? + i/ = 7i. 
{ xy = o5. 


15. 


{x + y + x^-\-y^=\S. 
1 xy = 6. 


^- j(y_i)^.2_3^^2. 


16. 


{4(x + y)^3xy. 

1 X + y -\- x^-\-i/= 26. 


1 0^-7/2= 175. 


17. 


(a^+ ;/ ,, 337. 
\x-^y = 7. 



(.,5+ 5 7/2 = 6 a;. ix^ + xy-\-x=\4. 

\x^-5y^ = 4:xy. ^^' \ y^ -{- xy + y = 2S. 

^ r 2:2 4. ^y^ 140. ^^^ {2^-yS = 20S. 

' \y'^ + xy = 06. " I xy(x — y) =z48. 

10. i^-^r!- 20. (-?/ + =^y = 12- 



xy-y' = 4. ■ \y + sc>y=lS. 



352 
21. 



ELEMENTS OF ALGEBRA. 



22. 



^ -\- if' ^=^'^xy. 
X + y = b. 

2xy +12 = Zx\ 
6 xy -{- 12 = a;^ 

2 3 

23. ^ ^ y 
lxy = 2. 

x-\- y 



,./: 



^4. 



24. 



25. 



26. 



27. 



29. 



30. 



31. 



32. 



l-\- xy 

I - xy 

2^4 + a;V + y/* = 7371. 
x^ — xy -\-f = 63. 

x^ + y^ = 641. 
0^2/(^2 + 2/^) =- 290. 

x^ -\- 3 a; ?/ + ;?/ = 19. 
a;2 + ;^2 ^ lo; 

a;2 — 3 xy + ;/^.= — 5. 
3^2_5^y+3y2^9 

y^ — x^ = a^. 
y — X = a. 

x^-\-y^ = Ux^7/. 
X + y = a. 

96 — x^y^ = 4:xy. 
X + y = 6. 

x^ — y/ = 56. 
x^^- xy \f-^ 28, 



33. 

34. 

35. 
36. 

37. 
38. 
39. 
40. 
41. 
42. 
43. 
44. 



y'^ — xy = a? ■{■ W'. 
xy — x'^ = 2 ah, 

x^y — y — 21. 
x^ y — X y — 6. 

^ 2 x^ 480 
2/2 + "^ - "49" • 
(^a;2 + 2/^ = 65. 

1 1 a; + ?/ 
X y 6 

a; + V 5 



/ 



x -\- y -[- I 



xP'y'^ + 5 a;?/ = 84. 
a; + ?/ = 8. 



■'« + y + Va: + y = 12. 
x^ + f^ 41. 



a; + ?/ + ^/x + y = 12. 
x^ + y'^ = 189. 

a; + Vxy + ?/ =: 19. 
x^ + xy + y^ = 133. 

a/^ + Vy = 4. 
V^3 -^ ^f = 28. 

a:?^ 4- 2/5 = 6. 
a:l + 7/1 = 126. 

?/2 — a;2 = 4 a &. 
xy = a^ — h'^. 

2x?- ?yxy + ?y2 = 4. 
2a;2/- 3 2/2-:i;2^-9. 




SIMULTANEOUS QUADRATIC EQUATIONS. 353 

48. <i^ 2^ 

La; y 

X -^ y xj-_y _ 10 

49. -{ x — y X + y~ 'S ' 
y + / = 45. 

X •\- y ^ x — y 

50. 'ia-^b~a — b 
xy = ab. 



51. The sum of the squares of the digits composing a 
number of two places of figures is 25, and the product of 
the digits is 12. Find the number. 

52. There are two numbers whose sum, multiplied by 
the greater, gives 144, and whose difference, multiplied by 
the less, gives 14. Find the numbers. 

53. The sum of the squares of two numbers is a, and 
the difference of their squares is b. Find the numbers. 
Solve for a = 170 and 6 = 72. 

54. A number divided by the product of its two digits 
gives the quotient 2 ; and if 27 be added to the number, 
the digits are reversed. Find the number. 

55. The sum of two numbers is a, and the sum of their 4th 
powers is b. Find the number. Solve for a = 4 and b = 82. 

56. The difference of two numbers is a, and the differ- 
ence of their cubes is 7 a' Find the numbers. 

57. The difference of two numbers is 3, and the difiTer- 
ence of their 5th powers is 3093. Find the numbers. 



354 ELEMENTS OF ALGEBRA. 

58. A number consisting of two digits has one decimal 
place ; the difference of the squares of the digits is 20, and 
if the digits be reversed, the sum of the two numbers is 11. 
Find the number. 

59. Find three numbers whose sum is 38, such that the 
difference of the first and second shall exceed the difference 
of the second and third by 7, and the sum of whose squares 
is 634. 

60. The small wheel of a bicycle makes 135 revolutions 
more than the larger wheel in a distance of 260 yards ; if 
the circumference of each were one foot more, the small 
wheel would make 27 revolutions more than the large 
wheel in a distance of 70 yards. Find the number of feet 
in the circumference of each wheel. 

61. Find two numbers such that their difference shall 
be a, and the product of their n\k\. roots c. Solve for a = 4, 
c = 2, and n = h. 

62. Find a fraction such if the numerator be increased 
and the denominator diminished by 2, the result will be its 
reciprocal; while if the numerator be diminished and the 
denominator increased by 2, the result will be \% less than 
its reciprocal. 

63. A principal of $10,000 amounts, with simple in- 
terest, to $14,200 after a certain number of years. Had 
the rate of interest been 1 % higher and the time 1 year 
longer, it would have amounted to $15,600. Find the 
time and rate. 

64. A sum of money at interest amounted at the end of 
the year to $10,920. If the rate of interest had been 1 % 
less, and the principal $100 more, the amount would have 
been the same. Find the principal and rate of interest. 



INDETERMINATE EQUATIONS. 355 



CHAPTER XXIV. 
INDETERMINATE EQUATIONS. 

147. Simple Indeterminate Equations are equations of 
the first degree that admit uf aii unlimited number of 
solutions. 

Thus, in 3x-2y = 2, if y = 2, x = 2; if 3/ = 3, a; = 2f ; if y = 5, 
X = 4; if y = Si X = 6 ; etc. It is evident that an unHmited num- 
ber of values may be given to y and x that will satisfy the equation. 
Hence, an ecj nation containing two unknown numbers admits of as 
many solutions as we please, and is indeterminate. 

Since the values of the unknown numbers are dependent upon 
each other, they may be confined to a particular limit ; as, for exam- 
ple, suppose the variables to be restricted to positive or negative inte- 
gers, we may thus limit the number of solutions. 

Example 1. Solve 19x-i-by= 119, in positive integers. 
Solution. Transpose 19 a:, 5i/=119— 19 2:. 

Therefore, y = 23-3x + 4l-^] (1) 

l-x 
Since the value of y is to be integral, then - must be integral, 

although fractional in form; and so also is any multiple of it. 

Let — r— = n, an integer. 

Therefore, x=l-5n (2) 

Substitute in (1), y = 20+19 n (3) 

We must take only such integral values for n as will give positive 

integral values for x and y. 

(2) shows that n may be 0, or have any negative integral value, 

but cannot have a positive integral value. 



356 ELEMENTS OF ALGEBRA. 

(3) shows that n may be and — 1, but cannot have a negative 
integral value greater than 1 . 

Therefore, n may be and — 1. 

^^"^^'^=2i}'^^'%=i}' 

Query. Can n be — 2 or + 1 ? Why ? 

Example 2. Solve 7 x — l^y = 19, in positive integers. 

Process. Transpose and solve for a;, x = 2-hy + bl — ;=— J (I) 

Let —j — = n, an integer. 

Therefore, y = 7n-l (2) 

Substitute in (1), x=l2 7i-\-l (3) 

Evidently x and y will both be positive integers if n have any 
positive integral value. 

Hence, x = 13, 25, 37, 49, ... . 

y= 6, 13,20, 27, .... 

Notes : 1. Having obtained a few of the possible values of x and y, the law 
will become evident. 

2. It will be seen from the above solutions that when only positive integral 
values are required, the number of solutions will be limited or unlimited ac- 
cording as the sign connecting the terms is positive or negative. 

Example 3. Solve I90x — 23y = 708, in least positive integers. 

/3 - x\ 
Process. Solve for y, y = Sx~30 — 6\—^j (1) 

S — x 

Let ^ = n, an mteger. 

Therefore, x = 3-2Sn (2) 

Substitute in (1), y = -6-190 n (3) 

Evidently x and y will both be least positive integers if n be — 1 . 
Therefore, n = — ], x = 26, and y = 184. 

Note 3. If the coefficient of the unknown number in the numerator of the 
fraction is not 1, it will be necessary to make several transformations. 

Example 4. Solve 21a:+ 17 y = 2000, in positive integers. 



» ._ — ", ail xiiiegei. 
3-n 


(2) 


3-»j 

— 7 — = w, an integer. 




n = 3 - 4 m. 




x= 17m-10 


(3) 


3/= 130- 21 m 


(4) 



INDETERMINATE EQUATIONS. 357 

11 -4x 
Solution. Solve for y, y = 117 - x + — r= — (1) 

w U-4x 
Transpose, y + a: — 117 = — yj 

Since x and y are to be integral, y + x — 111 will be integral ; 

11 — 4 a: .,, , , 

hence, — r^ — will be integral. 

ll-4a: 
Let 

Therefore, 

>.x 3-n , . 

Now —z — mwtt be integral. 

Let 

Therefore, 

Substitnte in (2), 

Substitute in (1), 

(3) shows that m may have any positive integral value, but can- 
not be 0, or have any negative integral value. 

(4) shows that m may have any integral value from to 6, or any 
negative integral value, but cannot have a positive integral value 
greater than 6. 

Therefore, m may be 1, 2, 3, 4, 5, Q, giving the following pairs of 
values : 

x= 7, 24, 41, 58, 75, 92. 

y = 109, 88, 67, 46, 25, 4. Hence, 

To Solve a Simple Indeterminate Equation, Involving Two 
Unknown Numbers, for Integral Values. Find tlie value of 
one of the unknown numbers. Place the fractional part of this value 
efjual to Uj an integer, and solve the resulting equation for the other 
unknown number. Substitute this result in the value first obtained. 
Solve the two simple equations thus fonuerl, by inspection, for inte- 
gral values of n. 

Notes: 4. It is better, in solving the original equation, to solve for the 
unknown number which has the least coefficient. 

5. A little insrenuity in arranging the terms will often obviate the necessity 
of a second transformation. 



358 ELEMENTS OF ALGEBRA. 

148. There can be no integral values of x and y in an 
equation of the form ax ±h y = c, ii a aiid b have a com- 
m6n factor not common also to c. 

For, suppose d to, be any factor of a and also of b, but not of c, 

sucli that a — md and h = n d. 

c 
Then mdx ±.ndy = c, or mx ztny = -j. 

Since m and n are integers, if 'x and y be also integers, mx ±. ny 

is an integer. But ^ is a fraction. Hence, no integral values of 

X and y can be found. 

Notes : 1. If a, b, and c have a common factor, it should he removed by 
division, then proceed as in Art. 147. 

2. The solution of any indeterminate equation of the form ax — by =. ±c, 
in which a and h are prime to each other, is always possible, and admits of an 
unlimited number of integral solutions (Ex. 2, Art. 147). If the equation be 
of the form ax-\- hy = c, the number of results Avill always be limited ; and, 
in some cases, the solution is impossible (Ex. 1, Art. 147). 

Exercise 130. 

Solve in positive integers : 

1. 2x-\-3y = 2o; Ux = 5y-7; 3x = 8y-16, 

2. 5r?:+ ll2/ = 254; 9 a: + 13 2/ = 2000. 

3. 15x-l7y=l; 13x-9y=:l; 9x-Uy=:10. 
Solve in least positive integers : 

4 3^ + 7y = 39; 3x+4y=:39; 7x+lDy = 22^. 

5. 27a;-192/ = 43; 2^ + 7?/ =125; 555y-22a: = 73. 

6. 19 ^-5?/ =119; I7x = 4:9y-S. 

Are integral solutions possible for the following? Why? 

7. 3^ + 21 7/ = 1000; 7 a; +14?/ = 71. 

8. 323 a?- 527 y= 1000; 166 a: - 192 y = 91. 



INDETERMINATE EQUATIONS. 369 

9. Solve 7 a- 4- 15 y = 145, in positive integers, so that 
X may be a multiple ot y. 

^ 146 , 145 n 

Suggestion. Let x-ny, then y = ^ ^^ , and x = ^^^^^ . 

10. Solve 'S9 X — 6 y = 12, in positive integers, so that 
y may be a multiple of z, 

11. Solve 20 a: — 31 2/ = 7, so that x and y may be 
positive, and their sum an integer. 

Suggestion. Put x + y = n. 

149. A problem is indeterminate when it involves less 
conditions than there are unknown numbers. 

Exercise 131. 

1. Find a number which being divided by 3, 4, and 5, 
gives the remainders 2, 3, and 4, respectively. 

Solution. Let x represent the number and y the sum of the 
quotients, then, 

x-2 a--3 x-4 
3 +^ + — = »• 

f'S-\-y\ 
Simplify and solve for ar, a- = y + 2 4- 13 I — j;^ I (1) 

3-l-w \ •*' / 

Let -^=- = n, an integer. 

Therefore, y = 47 n — 3. 

Substitute in (1). a: = 60 n - 1. 

Hence, n may be 1, 2, 3, 4, etc. 

Therefore, x = 59, 119, 179, 239, etc. 

.7 = 44, 91, 138, 185, etc. 

2. Find the least number which being divided by 2, 3, 
4, 5, and G, gives remaindei-s 1, 2, 3, 4, and 5, respectively. 

3. Find two numbers which, multiplied respectively by 
14 and 18, have for the sum of their products 200. 



360 ELEMENTS OF ALGEBRA. 

4. Divide 142 into two parts, one of which is divisible 
by 9, and the other by 14. 

5. There are two unequal rods, one 5 feet long and the 
other 7. How many of each can be taken to make up a 
length of 123 feet ? 

6. Find two fractions having 5 and 7 for denominators, 
and whose sum is ||. 

7. Find the least number that when divided by 9 and 
17 will give remainders 5 and 12, respectively. 

N- 5 
Suggestion. Let N represent the number, — - — = x, and 

N~ 12 

— p^ — = y. .'. 9x=l7y-{-1. 

8. A farmer bought sheep, pigs, and hens. The whole 
number bought is 125, and the whole price, $225. The 
sheep cost $5, the pigs $2.50, and the hens 25 cents. 
How many of each did he buy ? 



Solution. Let 






X = the number of sheep, 
y = the number of pigs, 


and 






z = the number of hens. 


Then, 




x + 


y + z= 126 (1) 


and 


5x+2. 


■5y + 


.25 z = 225 (2) 


From (1) and (2), 






?/=86-2x-^ (3) 


Let 






x-1 
Q - n, an integer. 


Therefore, 






x = 9n-\-l. 


Substitute in (3), 






y = M-l9n. 


Substitute in (1), 






2;= 40+ 10 n. 



Therefore, n may be 0, 1, 2, 3, and 4, giving the following values : 

x= I, 10, 19,28,37. 
y = 84, 65, 46, 27, 8. 
z = 40, 50, 60, 70, 80. 



PROBLEMS. 361 

Qaeries. How many solutions ? In how many diflFerent ways 
may the stock l)e bought? How solve by means of only two un- 
known numbers? 

9. How can one pay a sum of $ 1.50 with 3 and 5 cent 
pieces ? In how many ways can the sum be paid ? 

10. Can a grocer put up the worth of S3.50 in 11 and 
7 cent sugar ? In how many ways can he do it in even 
and odd pounds, respectively ? Find the greatest and least 
number of pounds of the 7- cent sugar he can use. 

11. Is it possible to pay £50 by means of guineas and 
three-shilling pieces only ? 

12. A owes B $5.15. A has only 50-cent pieces and B 
only 3-cent pieces. How may they settle the account ? 

13. A farmer bought horses at S 60 a head and sheep at 
S8, and found that he bad invested S4 more in sheep than 
horses. How many of each kind did be buy ? 

14. A farmer invested $1000 in 75 head of cattle, worth 
$25, $15, and $10 per head. Find the number of each 
kind, and the number of ways in wliich he could buy 
them. 

15. A grocer had an order for 75 pounds of t€a at 55 
cents a pound, but having none at that price he mixed 
some at 30 cents, some at 45 cents, and some at 80 
cents. How much of each kind did he use, and in how 
many ways can he mix it? 

16. How many pounds of 20, 35, and 40 cent coffee 
must a grocer take to make a mixture of 150 pounds worth 
30 cents a pound ? In how many ways can the mixture 
be made ? 



362 ELEMENTS OF ALGEBRA. 

17. How many gallons of S 1.50, S1.90, and $1.20 wine 
must a vintner take to make a mixture of 40 gallons worth 
$1.60 per gallon ? How many ways may the mixture be 
made ? Can an odd number of gallons of each kind be 
taken ? An even number ? 

18. In how many ways can £1 be paid in half-crowns, 
shillings, and sixpence, the number of coins in each pay- 
ment being 18 ? 

19. A hardware merchant paid $180 for 20 stoves. 
There were three sizes: one $19 each, another $7, the 
other $6. How many of each size did he buy? 

20. A person having a basket of oranges, containing 
between 50 and 72, takes them out 4 at a time, and finds 1 
over; he then takes them out 3 at a time, and finds none 
over. How many had he ? 

A^-l N 

Suggestion. Let N represent the number, — j— = x, and -« = t/. 

l+x ^ I + x 
.'. y = x+ ~^' Pwt — ^ = n. Then n must be 5 or 6. 

21. A poultry dealer has a basket containing between 
200 and 300 eggs, he finds that when he sells them 13 at a 
time there are 9 over, but when he sells them 17 at a time 
there are 14 over. Find the number of egfjs. 

22. Two countrymen together have 100 eggs. If the 
first counts his by eights and the second his by tens, there 
is a surplus of 7 in each case. How many eggs has each ? 

23. A surveyor has three ranging poles of lengths 7 feet, 
10 feet, and 12 feet. How may he take 40 of tliem to 
measure 113 yards? In how many ways may the mea- 
surement be made ? 



INEQUALITIES. 363 

CHAPTER XXV. 
INEQUALITIES. 

150. Since a positive number is greater than any negative num- 
l)er, the statement that a is algebraically greater than 6, or that a—h 
is positive, is expressed by a > 6 ; that a is algebraically less than 6, 
or that a — 6 is negative, is expressed by a < 6. Hence, 

An Inequality is a statement that on^ expression is 
greater or less than another; as, 

1 — ar > = — : m — n < x. 

The expression at the left of the sign is calle<l the first member, 
and the expression at the right, the second member of the in- 
equality. 

The fonn a> h> c, means that 6 is less than a but greater 
than c. 

Notes : 1. Inequalities are said to subsist in the same sense wlien the lirst 
member is the greater in each, or the first member is the less in each ; as, 
3 > 2, 7 > 5, and .5 > 3 ; a<h, c<d, and m<n. 

2. Two inequalities are said to subsist in a contrary sense when the first 
member is the greater in one, and the less in the other ; as, 5 > 3 and o < 6 ; 
m < 5 and h > n. 

3. An inequality is said to be solved when the limit to the value of the 
unknown numl)er is found. 

151. Subtract a + & from each member of a > 6, 

then, a - (a + b) > b- (a -\- 6). 

Simplify, . — 6 > — a, 

or, ~ a < — b Hence, 

I. If each member of an inequality has its sign changed, the sign of 
inequality will be reversed. 



364 ELEMENTS OF ALGEBRA. 



Multiply each member of 


- 5 < 5 by - 


-2, 


then, 


10 > - 10. 




Multiply each member of 


a > 6 by - 


-m, 


then, 


— am < — 6m. 




Divide each member of 


- 6 < 4 by - 


-2, 


then, 


3 > - 2. 




Divide each member of 


a > ft by - 


-m, 


then. 


a b 

m m' 


He 



II. If each member of an inequality be multiplied or divided by 
the same negative number^ the inequality tvill be reversed. 

Suppose a> b, c > d, m'> n, — 

By definition, a — b, c-d, m-n, are positive. 

Add, (^a -b) + {c - d) + (m - n) + .... is positive. 

or, (a + c + m -I .) - (b + d -\- n -\- ... .) is positive. 

Therefore (by definition), a + c -^ m + — > b + d + n + .,», 
Thus, 7 > 3 

5> 2 

4> 1 



Add, 16 > 6, or divide by 2, 8 > 3. Hence, 

III. If the corresponding members of several inequalities be added, 
the sum of the greater members will exceed the sum of the lesser 
members. 

Suppose a > b and m > n, then a - b and m — n are positive. 
But, (a — b) — (m — n), or (a — m) — (6 — n) may be either 
positive, negative, or 0. 

Therefore, a — m > b — n, a — m < b — n, or a — m = b — n. 



Thus, 5 > 3 

3> 2 



Subtract, 2 > 1 



7>4 
5> 1 

Subtract, 2 < 3 



8> 7 
6>5 

Subtract, 2 = 2. Hence, 



IV. If the members of one inequality be subtracted from the corres- 
ponding members of another, tht resulting inequality will not always 
subsist in the same sense. 



INEQUALITIES. 865 

1 2 X 3 z 64 

Example 1. Solve 3^ x ^ — > -^^ + ^^ for the limits of x. 

Solution. Free from fractions and simplify, 

112a;-6>45x+ 128. 
Subtract 45 x - 6, 67 a; > 134. 

Divide by 67, x > 2. 

Therefore, x is greater than 2. 

Example. 2. Solve the following : 

\bx-Zy>Zx-^b (1) 

i3a: + t/=r22 (2) 

Solution. Subtract 3 x from (1), 2 x - 3 y > 5 (3) 

Multiply (2) by 3, 9x + 3 y = 66 (4) 

Add (3) and (4), 11 x > 71. 

Divide by 11, x > Q^^. 

22 -w 
From (4), x = ^ • 

Substitute in (3) and simplify, —y>-\\. 

Therefore, 2/ < 2^^ (see I) 

Example 3. Solve the inequalities : 

{fix — mn^n^ — mx (1) 
\ mx ~ nx + mn < in^ (2) 

Process. Simplify (1) and solve, x > n. 

Simplify (2) and solve, x <rr. 

Hence, x is greater than n and less than m. 

Note. The principles applied to the solutions of equations may be applied 
to inequalities, except that if each member of an inequality haa iti sign 
changed, the lign of inequality will be reversed. 



Exercise 132. 
Find the limit of x in the following : 
1. 4x-3 >fa;-f; f — | ^ < 9 - 3 a:. 
^o o o 1 x^ — a a — X 2x a 



366 ELEMENTS OF ALGEBRA. 

o ax — 2b a X — a ax 2 
^- Zh 2h~ ^ 1" ~ 3 * 

4. If 2^2 + 4 a; > 12, show that x>2. 

5. If 7 a;2 - 3 a; < 160, show that x < o. 

6. If 4 ic + 12 — ic^ > 0, show that x is included be- 
tween 6 and — 2. 

7. If 9 a; < 20 it^ + 1, show that x > \ oi < \. 

8. If 15 — ic — 2 ic^ > 0, show that x lies between | 
and — 3. 



Q I ^ ;^ > 30 — 4 ic. ^Q ( 1 J ^ < I ^ + 3|-. 

< 3 :?: + 49. * ( 6 oj > 24 - 2 ^. 



Find an integral value of x in the following 

\ IQx 

( |(^ + 2) + ^ < I (^ - 4) + 9. 
' \l(x-\-2) + \x>l{x+l)^^. 

Find the limits of x and y in the following : 

.o i3^4-5^>121. |7.^; + 5^>19. 

* (4a;+ 72/ =168. / a; - 3/ = 1. 

(« + &) ^ — (a — ?)) 2/ > 4 a &. 



^ I (tt - &) ^ + (a 4- 



h)y = 2(a + h) {a - 6). 



15. A certain number plus 5, is greater than one third 
the number plus 55 ; while its half plus 2, is less than 41. 
Find the number. 



INEQUALITIES. 367 

16. Find the price of oranges per dozen, when three 
times the price of one orange, decreased by tiiree cents, is 
more than twice its price increased by one cent ; and eight 
times the price of one orange, decreased by twenty cents; 
is less than three times its price increased by ten cents. 

152. Since the stjuare of a negative number is positive, if a and b 
represent any two numbers, (a — 6)^ must be positive, whatever the 
values of a, and b. Therefore, since every positive number is greater 
than zero, 

(a - 6)2 > 0. 

Expand, a^ - ^ah + h'^> 0. 

Add 2 a 6 to each member, a* + 6^ > 2 a 6. Hence, 

The sum of the squares of two unequal numbers is greater 
than tvnce their product. 

Vote. The above is a fundamental principle in inequalities. 

Example 1. Show that a"^ -\- 1)^ ■\- c^ > ab + a c + h c, a and h 
positive. 

Prool Since a, b, and c are any unequal numbers, 

a2+62>2a6 (1) 

a«+c«>2ac (2) 

62 + c«>26c (3) 

Add the corresponding members of (1), (2), and (3), 

2aa + 2 62-l-2c2>2a6+2ac + 2 6c. 

Divide by 2, a« + 6« + c^ > a 6 + a c + 6c. 

Query. How if a = 6 = c ? 

Example 2. Show that a« + 6« > a* 6 -}- a 6*. 

Proof. We shall have, a* + }fi > a'^b + a 6*. 

Factor, (a + 6) (a« - a 6 + 6^) > a 6 (a + 6). 

Divide by a + 6, a'^ — ah -\- b"^ > ab. 

Add a 6, a« + 6^ > 2 a 6. 

Therefore, a« + 6» > a^ 6 + a 6*. 



368 ELEMENTS OF ALGEBRA. 

Example 3. Which is the greater, V/ 1- 1/ — r- or /y/a i 

Proof. We shall have, 

Square each member, 
a2 62 



2 Y a bmn -\ r- > or < a 6 + 2 \/a bmn + m n. 

it9f tV CI O 

, a^h^ m^n^ 

Subtract 2 \/ahmn^ 1- — r- > or < ao + ww. 

^ m n a 

Free from fractions and factor, 

{ah + m n) {a^ h^ — abmn -\- m^ n^) > or < abmn(ah + mn). 

Divide by ab -\- mn, 

a^h^ — abmn + m^n^ > or < ab mn. 

Add a b 7« n, a^ b^ + m^ n^ > or < 2 a 6 m n. 

But, a^b^-i m'^n^':> 2abmn. 

Therefore, \ 1^ + \ -afT > ^""^ -^ V^""- 



Exercise 133. 

Show that, the letters being unequal, positive, and 
integral : 

h^ a^ a b 

2. a 6-2 + a-% > a-^+ b-^ ; (ni^-]- n^) (mH n^) > (m^+ n^f. 

3. xy-\-xz-\-yz < {x + y^z)^-\-{x+z—yf'-{-{y + z~x)\ 
Which is the greater : 

. „ . „ , a + b 2 ab m n 11 

4. 71^4-1 or ?i2+ ^ ; —^ or —-7 ; -5 s or 

5. 1±J or ^^; 3(1 + a2 + a^) or (1 + c^ + a'f, 

ga^ y x^ — y^ ^ 



INEQUALITIES. 369 

1/9 a/^ 
V3 v5 
Queries. How in 4 and 6, if a = 6 ? In 4, if n = 1 ? 

7. If a^ + ^>2 + c2 = 1, and a?» + y» + «2 = 1, show that 
aa; + &y + C2 < 1. 

Query. How ifa = 6 = c = a: = y = z? 

If a > 6, show that : 

8. a - 6 > iVa - V6)^ a^ + 7 aH > (rr + &)» 

9. a-6* > a*6« ; a^ + 13 a ^2 > 5 a26 + U 63. 

Miscellaneous Exercise 134. 
Example 1. Solve the inequahties : 



{: 



V2(xy-f-y2) + 4<;y(2i/-l)(y + a:) (1) 
2x + 5y>8 (2) 

Solution. Square each member of (I) and simplify, 
2xy-\-2y^-\- 4 <2y^-^2xy -y-x. 
Subtract 2ari/ + 2 y«, 4 <-y-x (3) 

Multiply each member of (3) by 2, 

8<-2y-2a: (4) 

Add the corresponding members of (2) and (4), 

3y > 16. .-. y >5f 
Multiply each member of (3) by 5, 

20<-5y-5a: (5) 

Add (2) and (5), - 3a: > 28, or 3a: < -28. .-. x < -9f 

Example 2. Simplify (y-\-x<m — n) (m* + m n 4- 71^ > y — x). 

Solution. We are to multiply the corresponding members 
together, (y + x) (y — x) = if — a:*, 

(m — n) (m* ■\- mn -\- n'-*) = m* — n*. 
Therefore, (y +x < m-n){mHmn-\- n^ >y-x) = y^~x^ < m*- n». 

24 



370 ELEMENTS OF ALGEBRA. 

Example 3. Which is the greater, x^ + y^ or x^y -\- y*xl 

Proof. We shall have, x^ + y^> or <: x* y -{- y* x. 

Subtract x^y -\- y^x, x^ — x^ y + y^ — y* x > or < 0. 

Factor, (x^ - y^) (x-y)> or < 0. 

Now, whether a; > or < .y, the two factors, x^ — y^ and x — y, will 
have the same sign. Hence, since (x'^—y^) i^~y) is always positive, 

{x^ - y^) {x-y)> 0. 
Therefore, a;^ + 2/^ > x^ y -\- y^x. 

Example 4. Which is the greater, m* — n* or 4 n^ (^ _ ,^) when 
m > n ? 

Proof. We shall have, m^ — n* > or < Am\m—n). 

Divide by w — n, m^ + wi*n + m w^ + n^ > or < 4 m^. 

Subtract wi^ + m^ n and factor the resulting inequality, 

n^ (m + n) > or < m\^m—n). 
But, m > w (1) 

Square (1), m^ > n^ (2) 

Multiply (1) by 2, 2 m > 2 n. 

Add m — n, 3m — ??>m + n (3) 

Multiply the corresponding members of (2) and (3), 

m^ (3 m — n) > n' (m -{- n). 
Therefore, 4m^ (m — n) > w* — n*. 

6. Find the sura of x^ + y > 1 — a, y^ — 2 a > 5 + 4, 
^ X + y < 2 a + 1, and y^ - S x^ < 5 - a. 

6. From a^ + 2 a a;^ < 5 take a (a + x^) y n^ —1. 

7. From a2 < 3 - .7.2 subtract 2 ^2 > 5. 

Multiply : 

8. {a + hf > {x-yf by -3; Z-f < 5-^^ by x^ + y\ 

9. Divide a^-l^ > a^+ h^ by a2 + ^2 
10. Divide 11 a2 + 88& > 121^2 ^y - H. 



INEQUALITIES. 371 

Perform the indicated operations and simplify : 

11. (w-l< 5)(m + l<10); (a < n + h){n-b > c). 



12. (_ 2 > - 3)3 ; (5 > 2) -f- (3 < 4) ; V25 > 9. 

13. [- 243 > - 1024]i ; (71 + 1)^ > n^ - n^ + 4: n. 

14. m3 - 7i3 > (m - n) {m^ + n^) ; 4^- 64 < 8. 

15. (m2 - n^ < u.^)'T-{ix> m + n); [- ?t > i/f. 

Solve : 

16. {X''2f > 0^+ 6 x-25',V(x-l)^ + 'S 2^ + 6 >2xK 

17. a; - 2 > V ^-^=^ ; V3 - 4 Vi > VI6 2: - 5. 



^g |3y+2a;>3. ^^ | a: + f > Vo^ - 3 a; + y. 
( 4 > 4 7/ + ic. ■ ( 5 > a: — y. 

j42/-a^>?/ + 4. J3:r-l>a: + 3y. 

^"- (3a;-6y> l-4i/. "^ ( 27/-3rr2 = 3a;-3.x2. 

22. 38a:-7-15a:2<0; 6 ar^ 4- 7 ;r + 2 < 0. 

23. 17 a;- 6r»-5<0; 6 a: + 11 - a:^ < 2 a; - 10. 
Find integral values of x in the following : 

(3ix-.5x>5. {x + 7 

^^- l2.5a: + ia;<8. ^^' (2a; + 



a; + 7 < 15. 

10 > 20. 



26 U^-i^<3. 27 |2a:-5>31. 

I 7a;-15>4a;+30. ^' (3a;-20<2ar. 

28. ar2 4- 2a;-15<0; a:^ ^ lOa: 4. 63 < 0. 



372 ELEMENTS OF ALGEBRA. 

Show that: 

29. Vl9+V3> VIO + VT; V5 + Vn > V3+3V2c 

li a >h, show that : 



30. Va^ - 62 + Va^ - (a - hf > a. 

31. a^-h^ <3a^{a- h) and >Sb^{a- b). 

32. a-h> -j-^ and < -^3- . 

If x^ — a^ + y^, if' — (p- -^ d'^^ show that: 

33. xy'^ac + bdoTad + bc. 
Show that: 

34:. {a b + X I/) (a X + b y) ^ 4: a b xy, 

35. (ti + 6) (ft + c) (6 + c) > 8 a & c. 

36. Show that the sum of any fraction and its reciprocal 
is greater than 2. 

?7. In how many ways may a street 20 yards long and 
15 wide be paved with two kinds of stones ; one kind 
being 3f feet long and 3 wide, the other 4| feet long and 
4 wide ? 

38. A and B set out at the same time to meet each 
other; on meeting it appeared that A had travelled a miles 
more than B, and that A could have gone B's distance 
in n hours, and B could have gone A's distance in m 
hours. Find the distance between the two places. Solve 
when ft = 18, ^ = 378, and m = 672. 



SERIES. 373 

CHAPTER XXVI. 
SERIES. 

153. A Series is an expression in which the successive 
terms are formed according to some fixed law ; 

As, 1, 2, 4, 8, , in which each term is double the preceding 

term ; a, a + c/, a + 2d, a + 3d, , in which each term exceeds 

the preceding term by d. 

ARITHMETICAL PROGRESSION. 

154. The expressions 1, 6, 9, 13, 17, , and 16, 10, 6, 0, 

— 5, —10, ...., are called arithmetical progressions or series. The 
first is an increasing series, and the second a decreasing series. 
The general form for such a series is, 

a, a + d, a-\-^d, a + 3<i, a + 4ci, a + Hd, a + 6cl, .... 

in which a is the first term and d the common difference ; the series 
will be increasing or decreasing according as d is positive or negative. 
Hence, 

An Arithmetical Progression is a series in which the adja- 
cent terms increase or decrease by a common difference. 

In every arithmetical series the following elements occur, any 
three of which being given, the other two may be found : 

The first term, or a. 

The last term, or /. 

The common difference, or d. 

The number of terms, or n. 

The sum of the terms, or s. 



374 ELEMENTS OF ALGEBRA. 

By an examination of the general form it is seen that the coefficient 
of d is always 1 less than the number of the term. 
Thus, the 2d term is « + rf, or a + (2 — 1) cf, 
3d term is a + 2 </, or a + (3 — 1) rf, 
4th term is a f- 3 </, or a + (4 — 1) 6?, 
12th term is a + 11 rf, or a + (12 — 1) d, and so on. 
In the nth, or last term, the coefficient of c? is n — 1. Hence, 

To Find the Last Term of an Arithmetical Series, when the 

first term, the common difference, and the number of terms 

are given. 

I = a+ {n-l)d (!) 

Note. The common difference may always be found by subtracting any 
term of the series from that which immediately follows it. 

Example 1. Find the 18th term of the series |, f, |, etc. 

Process. Here, n = 18, a = |, and d — ^ — l = \. 
Substitute these values in (i), / = | + (18 - 1) J = 4. 

Example 2. Find the 30th term of the series x -\- y, x, x — y, etc. 

Process. Here, n = 30, </ = x — {x+y) = —y, and a = x + y. 
Substitute these values in (i), I = x + y + (30 — l)(—y) = x — 28y. 



Exercise 135. 

Find: 

1. The 15th term of 7, 3, - 1, .... 

2. The 27th and 41st terms of 5, 11, 17, .... 

3. The 20th and 13th terms of - 3, - 2, - 1, .... 

4. The 37th and 89th terms of- 2.8, 0, 2.8, .... 

5. The 40th term of 2 a — h, 4 a - 3h, 6 a — 5h, .. 

6. The 15th and 8th terms of J, 1 . | 



ARITHMETICAL PROGRESSION. 375 

7. The first term is J, the 102d is 18. Find the com- 
mon difference. 

8. The 21st term is 53, and the common difference is 
— 2J. Find the first term. 

9. The first term is 5^, and the common difference is 3 J. 
What term will be 42 ? 

10. The first term is ^, the common difference is ||, and 
the last term is 17^. Find the number of terms. 

11. The 54th and 4th terms are - 125 and 0. Find 
the 42d term. 

12. Find three terms whose common difference is J, 
such that the product of the second and third exceeds that 
of the first and second by. 1 J. 

155. Taking the elements as given in Art. 154 : 

s = a + (a + rf)+(« + 2</) + (a + 3rf)+(a + 4f/) + ••.. ^ 
or 5 = / + (;-r/)-f (/-2c?H(/-3£/) + (/-4cO + .-.. a- 

Add, 2s = {a + [)-\-{a + l)-\-{a-\-l) + (a + l) + (a + l) -f .... to n terms, 
or 25 = n(a + /) (1) 

Substitute the value of I from (i) (Art 164) in (I), 
a » = n [2 a + (n — 1) d]. Hence (solve for s), 

To Find the Sum of all the Terms of an Arithmetical Seriei 

» = ^[aa-f (n-l)<f] (iH) 

Example 1. Find the sum of an arithmetical series of 17 terms, 
the first term being 5^, and the last term 25 i. 
Process. Here, n = 17, a = 5^, and / = 2.''>|. 
Substitute these values in (ii), « = y. (5^ + 25^) = 263j. 



376 ELEMENTS OF ALGEBRA. 

Example 2. Find the sum of the series 3|, 1,-1^, , to 19 

terms. 

Process. Here, n = 19, a = 3^, and c? = 1 — 3^ = — 2^. 
Substitute these values in (iii), 

s= ^-[2 X 3^ + (19 - 1)(- 2^)] = - 361. 

12 3 
Example 3. Find the sum of m , 3 m , 5 m , . . . . , to 

m' m m' ' 

m terms. 

Process. Here', n = m, a = m — — , and d =3m —~ ~ m — -) 

= 2m — - • 
m 

Substitute in (iii), 

2 m^ — m — 1 



=fK'"-9+^'" -'>("" -9] 



Example 4. The first term of a series is 3 m, the last — 35 m, and 
the sum — 320 7W. Find the number of terms and the common 
di Herence. 

Process. Here, s =i — 320 m, a = 3 m, and / = — 35 m. 
Substitute in (ii), 

- 320 m = ■x(S m - 35 m) = - 16 mn. .-. n = 20. 
Substitute in (iii), 

- 320 m = %0- [6 m + 19 rf] = 60 m + 190 c?. .-. d = -2m. 

Example 5. How many terms of the series — C|, — 6f , —6, , 

must be taken to make - 52| ? 

Process. Here, s= — 52|, a = — 6|, and c/ = f . 

n 
Substitute in (iii), - 52| = ^ C" ¥ + (n - 1) X f ]. 

Simplify and solve for n , n = 1 1 or 24. 

Query. Do both of these values satisfy the conditions ? In 
explanation write out 24 terms of the series and observe that the 
last 13 terms destroy each other. 



ARITHMETICAL PROGRESSION. 377 

Exercise 136. 

Find the sum of : 

1. 5, 9, 13, ..... to 19 terms. 

2. 10 J, 9, 7 J, ...., to 94 terms. 

3. 3 a, a, — a, . . . . , to a terms. 

4. 3 J, 2 J, 1|, ...., to n terms. 

^w— Im — 2 m — 3 

5. , , , ...., to m terms. 

m mm 

. 2ag~l ^ 3 6 g^ - 5 

6. , 4a , , ..... to n terms. 

a a a 

^ 4a + & 5a + 2& 

7. a, — X — , ^ , ...., to 19 terms. 

8. The first term is 3^, and the sum of 14 terms is 84J. 
Find the last term. 

9. The sum of 40 terms is 0, and the common difference 
is — ^. Find the first term. 

10. Find the number of terms and common difference: 

(1) when the sum is 24, the first term 9, and tlie last —6; 

(2) the sum 49 a, the first term a, and the last 13 a. 

11. The sum of 12 terms is 150, and the first is 5J. 
Determine the series. 

12. Show that the sum of the first n odd numbers is r?, 

13. Find the sum of all the odd numbers between 100 
and 200. 



378 ELEMENTS OF ALGEBRA. 

14. The sum of five terms is 15, and the difference of 
the squares of the extremes is 96. Find the terms. 

15. Find the sum of -=i, tj , 7=, ...., 

l'\- Vx ^-^ 1- Vx 
to n terms. 

156. a is called the arithmetical mean between a — d and a + d. 
Hence, 

An Arithmetical Mean is the middle term of three num- 
bers in arithmetical series. 

If a and 6 represent two numbers, and A their arithmetical mean, 
the common difference is A — a, or b — A. Therefore, 
A — a — h — A. Hence (solve for J.), 

To Find the Arithmetical Mean Between two Terms. 

A = ^ M 

If a and I represent any two numbers, and m the number of means 
between them, the whole number of terms is m + 2, or wi + 2 = n. 
Substitute this value for n in (i) (Art. 154), 

I = a + (m + I) d. Hence (solve for d), 
To Insert any Number of Arithmetical Means Between two 

'^«"-- ^ . 1- (V) 

This finds d, and the m required means are, 

a-\- d, a + 2d, a + ^d, a + 4Ld, ...., a-\-md. 

Example 1. Find the arithmetical mean between: (1) 27 and 

— 5 ; (2) rri^ -\- mn — n^ and m^ — m n -\- n^. 

Process. (1) Here, a = 27, 6 = — 5. 

27-5 
Substitute in (iv), A = — -x — = 11. 

(2) Here, a — m^ + mn- n^, b =m^ — mn + rfi. 

^ , . . ,. ^ , m^ + mn -- n^ + m^ — mn + n^ 
Substitute in (iv), A = 2 — =; m\ 



GEOMETlilCAL TUOGRESSION. 379 

Example 2. Insert five arithmetical means between 12 and 20. 
Process. Here, a = 12, / = 20, and m = 5. 

20- 12 
Substitute in (v), d = ^ ^ = H- 

The series is 12, 13^, 14|, 16, 17^, 18f, 20. 

Exercise 137. 
Insert : 

1. 14 arithmetical means between — 7-J and — 2^. 

2. 16 arithmetical means between 7.2 and — 6.4 

3. 10 arithmetical means between 5 m— 6 n and 5w— 6 m. 

4. 4 arithmetical means between — 1 and — 7. 

5. X arithmetical means between a^ and 1. 

« ^. , , . , . , , m— n , m-hn 

6. Find the arithmetical mean between — — - and . 

m+n m — n 

7. The arithmetical mean between two numbers is — 9, 
and the mean between four times the first and twelve times 
the second is — 66. Find the numbers. 

GEOMETRICAL PROGRESSION. 

157. The expressions 3, 9, 27, 81, ...., and 1, -i, i, -^, ...., 
are called geometrical progressions or series. The general form for 
such a series is, 

a, ar, ar^, ar*, ar*, ai*, an*, ar'', ...., 

In which a is the first term, and r a constant factor or ratio. Hence, 

A Geometrical Progression is a series in which the adja- 
cent terms increase or decrease by a constant factor. 

The Common Ratio is the fiictor by which each term is 
multiplied to form the next one. 



380 ELEMEJ^TS OF ALGEBRA. 

In every geometrical series the following elements occur; any 
three of which being given, the other two may be found. 

The first term, or a. 
The last term, or I. 
The common ratio, or r. 
The number of terms, or n. 
The sum of the terms, or s. 

By an examination of the general form it is seen that the expo- 
nent of r is always 1 less than the number of the term. 
Thus, the 2d term is a r, 
3d term is a r^, 
4th term is a r*, 
12th term is ar", and so on. 
In the nth, or last term, the exponent of r is w — 1. Hence, 

To Find the Last Term of a Geometrical Series, when the 

first term^ the comtnon ratio, and the number of terms are 

given. 

I = ar""-^ (i) 

Notes: 1. The common ratio is found by dividing any term by that which 
immediately precedes it. 

2. A geometrical series is said to be increasing or decreasing, according as 
the common ratio is greater than 1, or less than 1. 

3. An arithmetical series is formed by repeated addition or subtraction; a 
geometrical series by repeated multiplication. 

Example 1. Find the 8th term of the series .008, .04, .2, etc. 
Process. Here, a = .008, n = 8, and r = .04 ^ .008 = 5. 
Substitute in (i), / = .008 X 58-i = 625. 

Example 2. Find the 10th term of — , x, y, — , 

Process. Here, a = — , n = 10, and r = x -. — = - • 

y\ ' y X 

Substitute in (i), / = - f|j = ar-'yS 



GEOMETRICAL PROGRESSION. 381 

Exercise 138. 
Find: 

1. The 5th and 8th terms of 3, 6, 12, .... 

2. The 10th and 16th terms of 256, 128, 64, .... 

3. The 8th and 12th terms of 81, - 27, 9, .... 
4 The 14th and 7th terms of gJ^, ^^, 3^, .... 

.- rr.1 « 1 ^ X mx m^x 

5. The 6th term of -, — 5-, — 3- 

y / f 

6. The mth term of x, x^^ 7^, .... 

7. The 3d and 6th terms are f^ and — |. Find the 
series and the 12th term. 

8. The 5th and 9th terms are f J and §. Find the 
series. 

9. If from a line a inches in length, one third be cut 
off, then one third of the remainder, and so on; what part 
of it will remain when this has been done 5 times ? When 
/ times. 

168. Taking the elements as given in Art. 157, 

8z=a + ar-\-ar^ + ar*-\- -\-ai*-^-\-ar^'^ (1) 

Multiply (1) by r, 

sr = ar-i-ar^-i-aH»-| -f- a r"-'^ -I- g r»-^ -t- g r (2) 

Subtract (1) from (2), 

sr—8 = ar^ — a (3) 

Substitute the value of or* from (i) (Art. 167) in (3), and factor 
the result, « (r - 1) = r / - a. Hence (solve for <), 

To Find the Sum of all the Terms of a Geometrical Series 

8 = ^^null (ii) 

r - 1 



382 ELEMENTS OF ALGEBKA. 

Example 1. Find the 6th term and the sum of — J, ^, - f , . 
Process. Here, a = —^, 7i = 6, and r = — |. 
Substitute in (i) (Art. 157), ^ = - |^ x (- |)^ = |^. 

Substitute in (iii), 5 = " '^_^s _ ^ ^ = W- 

EXxiMPLE 2. Find the least term and the sum of 3, — 9, 27, 
to 7 terms. 

Process. Here, a = 3, n — 1, and r — — 3. 
Substitute in (i) (x\rt. 157), Z = 3 (- 3)« = 2187. 

3^:^y— =1641. 



Substitute in (ii), 


^= -3 




Exercise 139. 


Find the sum of: 




1. 3, -1, J,.... 


, to 6 terms. 


2. -lh-i>' 


,..., to 6 terms. 


3. 1, -J, A. •• 


. ., to 8 terms. 


1 3 
*• V3' ' V3' 


...., to 8 terms. 


5. 1, 3, 32, ..... 


to m terms. 


6. 2, -4, 8, ... 


. , to 2 m terms. 



7. The 7th and 4th terms are 625 and — 5. Find the 
1st term, and the sum of the 4th to the 7th terms inclusive. 

8. The sum of the first 10 terms is equal to 33 times 
the sum of the first 5 terms. Find the common ratio. 

9. The sum of three numbers in geometrical progression 
is 216, and the first term is 5. Find the common ratio 
and the numbers. 



GEOMETRICAL PliOGRESSION. 383 

159. A Geometrical Mean is the middle term of three 
numbers in geometrical series. 

If a and b represent two numbers, and G their geometrical mean, 

G b 
the common ratio is — , or ^. Therefore, 

G b 

— = jy. Hence (solve for G), 

To Find the Geometrical Mean Between two Terms 

O = \/ab ' (iv) 

If a and b represent any two numbers, and m the number of means 
between them, the whole number of terms is m + 2, or m -f- 2 = n. 
Substitute this value for n in (i) (Art. 167), 

/ = a r"* + 1. Hence (solve for r), 

To Insert any Number of Geometrical Means Between two 
Terms. , , _ 

This finds r, and the m required means are, 

ar, ar^, ar^, ar^, ar^ , ar»*. 

Example 1. Find the geometrical mean between : (1) — -= and 
3 V3 

7^; (2) 3x»y and I2xfz. 

"^^ 1 3 

Process. (1) Here, a = -—p, and b = —-z • 



Substitute in (iv), G = v/ — p ^ ^=^ 



3^ 

V3 V3 

(2) Here, a = 3x*y and b = Uxy'z. 

Substitute in (iv), G = ^33*y X I2xy*z = 6x^y^ ^z. 

Example 2. Insert six geometrical means between 14 and - ^y. 

Process. Here, a = 14, / = - /y, and m = 6. 

Substitute in (v), r = ^—^\^ = — i- 

Hence, the series is 14, -7, f - }, |, -^^h- ^. 



384 ELEMENTS OF ALGEBRA. 

Exercise 140. 

Find the geometrical mean between : 

1. 7 and 252; a^h and ah^-, f and |i ; | and ||. 

2. yV '^"cl jJ^o ; 4^:2 - 12a; + 9 and ^x^-\-12x^- 4. 
Insert : 

3. 2 geometrical means between 5 and 320. 

4. 2 geometrical means between 1 and \. 

5. 3 geometrical means between 100 and 2J|. 

6. 6 geometrical means between 14 and — -^^. 

7. 7 geometrical means between 2 and 13,122. 

8. Which is the greater, and how much greater, the 
arithmetical or geometrical mean between 1 and \. 

9. Find two numbers whose sum is 10, and whose geo- 
metrical mean is 4. 

HARMONICAL PROGRESSION. 

160. The expressions h ^, \, \, ...., and 4, -f, - f, -4, ...., 
are called harmonical progressions or series, because their reciprocals 

1, 3, 5, 7, , and ^, — |, — |, — |., . form arithmetical series. 

The general form for such a series is, 

I- ^> Wh^ ^TT^^ry-.- Hence, 

An Harmonical Progression is a series the reciprocals of 
whose terms form an arithmetical series. 



HARMONICAL PROGRESSION. 385 

Notes: 1. Evidently all questions relating to harmonical pro^^ression are 
readily solved by writing the reciprocals of the temis so as to form an arith- 
metical series. 

2. There is no general formula for finding the sum of the terras of a har- 
monical series. 

3. The term harmonical is derived from the fact that musical strings of 
•qual thickness and teiisiou produce harvumy when sounded together, if their 
lengths are represented by the reciprocals of the series of natural numbers; that 
is, by the series 1, J, J, i, J, J, etc. Harmonical properties are also interesting 
because of their importance in geometry. 

Example 1. Find the mth term of the series 3, IJ, 1, f , f , etc. 
Solution. Taking the reciprocals of the terms, we have ^, |, 1, 
|, |, etc. ; an arithmetical series. 
Here, a = |, rf = J, and n = m. 

Substituting in (i) (Art. 154), d = ^ + {m - I) ^ = ^. Taking 

the reciprocal of this value for the required term, we have — . 

Example 2. The 12th term is |, and the 19th term is ^. Find 
the series. 

Process. The 12th and 19th terms of the corresponding arith- 
metical series are 5 and ^. 

From (i) (Art. 154), 5 = o + 11 rf, 
^ = a + 18 d. 

Solving for a and rf, a = | and rf = J. 

The arithmetical series is, |, |, 2, t, |, 3, Y> • • • • 

The harmonical series is, |, |, ^, ^, |, J, i%, 

161. A Harmonical Mean is the middle term of three 
numbers in harmonical series. 

If a and b represent two numbers, and H their harmonical mean, 
the corresponding arithmetical series is -, ^, ^. The common dif- 

.11 !!,«,. 
ference is ^ ~ o* ^^ 6 ~ H' ^*^^'^'®^» 

D- — - = T - ^. Hen'ce (solve for //), 

25 



386 ELEMENTS OF ALGEBRA. 

To Find the Harmonical Mean Between two Numbers. 

H - ^^^ (i) 

Example L Find the harmonical mean between : (1) ^ and ^ ; 
(2) X -j- y and x — y. 

Process. (1) Here, a = \ and b — ^j^. 

Substitute in (i), H = \ 

(2) Here, a = x -\- y and b = x — y. 

X^ - 7/2 

Substitute in (i), H— '— . 



Example 2. Insert three harmonical means between f and ^^. 
Process. The terms of the corresponding arithmetical series are 
I and J^. 

Here, a = |, I = ^^, and m = 3. 
Substitute in (v) (Art. 156), d = ^. 
The three arithmetical means are ^, ^, ^. 
The required harmonical means are y\, f , ^. 



Exercise 141. 

1. Find the 8th term of IJ, l^f 2-^2^, .... 

2. Find the 21st term of 21 llf, 1^^, .... 

3. The 39th term is y^-, and the 54th term is ^. Find 
the series. 

4. The 2d term is 2, and the 31st term is ^*y. Find the 
first six terms. 

Insert : 

5. One harmonical mean between 1 and 13. 

6. 3 harmonical means between 2f and 12. 



HARMONICAL PROGRESSION. 387 

7. 4 harmoiiical iiu'iiii.s buLu uuii | iind ^. 

8. 6 harinouical means between 3 and ^. 

9. The arithmetical mean of two numbers is 9, and the 
harmonical mean is 8. Find the numbers. 

10. The difference of the arithmetical and harmonical 
means between two numbers is 1. Find the numbers; one 
being three times the other. 

11. Find two numbers such that the sum of their arith- 
metical, geometrical, and harmonical means is 9|, and the 
product of these means is 27. 

12. The arithmetical mean between two numbers ex- 
ceeds the geometrical by 2^, and the geometrical exceeds 
the harmonical by 2. Find the numbers. 

13. The sum of three terms of a harmonical series is 37, 
and the sum of their squares is 469. Find the numbers. 

14. The sum of three consecutive terms in harmonical 
series is 1^, and the first term is J. Find the numbers. 

15. Arrange the aritlimetical, geometrical, and harmoni- 
cal means between two numbers a and h in order of 
magnitude. 

16. If 50 potatoes are placed in a line 3 feet from each 
other, and the first is 3 feet from a basket, how far will a 
person travel, starting from the basket, to gather them up 
singly, and return with each to the basket ? 

17. There are four numbers in geometrical progression, 
the first of which is less than the fourth by 21, and the 
difference of the extremes divided by the difference of the 
means is equal to 3 J. Find the numbers. 



388 ELEMENTS OF ALGEBRA. 



CHAPTEK XXVII. 
RATIO AND PROPORTION. 

162. The Ratio of two numbers is their relative magni- 
tude, and is expressed by the fraction of which the first is 
the numerator and the second the denominator. 

Thus, the ratio of 10 to 5 is expressed by the fraction ^-^ ; the 
ratio of I to I is expressed by the fraction f -r f (= y%). 

The ratio of two quantities of the same kind is equal to the ratio 
of the two numbers by which they are expressed. 

Thus, the ratio of $5 to $6 is | ; of 15 apples to 3 apples is i/ ; of 
3f feet to 5^ feet is 3f ~ 5| = ^|. 

, The Sign of ratio is the colon :, -^, or the fractional 
form of indicating division. 

a 
Thus, the ratio of a to 6 is expressed by a : b, or a -f 6, or t, any 

one of which may be read "a is to 6/' or "ratio of a to &." 

The Terms of a ratio are the numbers compared. The 
first term is called the antecedent, the second the conse- 
quent, and the two terms together are called a couplet. 

A ratio is called a ratio of greater inequality, of less 
inequality, or of equality, according as tlie antecedent is 
greater than, less than, or equal to, the consequent. 

An Inverse Ratio is one in which the terms are inter- 
changed ; as, the ratio of 7 : 8 is the inverse of the ratio 
8:7. 

A Compound Ratio is the product of two or more simple 
ratios; as, the compound ratio 2 : 3, 5 : 4, 15 : 6, is 150 : 72. 



RATIO AND PROPORTION. 389 

NotM : 1. A quantity may be detiaed as a definite portion of any magni- 
tude. Thus, any definite number of dollars, poimds, bushels, acres, feet, yards, 
or miles, is a quantity. 

2. To compai*e two quantities they must be expressed iu terms of the same 
unit. Thus, the ratio of 2 rods to 9 inches is expressed by the fraction, 
16J X 2 X 12 396 



163. Evidently the ratios 4 : 5, 8 : 10, ^ : Yi are equal to each 
other. Ill general, 

a ma „ 

I. If the terms of a ratio are multiplied or divided by 
the sanu number, the value of the ratio is not changed. 

The ratio 9 : 7 is compared with the ratio 4 : 3 by comparing ^ 
and |. ^ = ^\, and | = ^|. Therefore, 4 : 3 is greater than 9 • 7. 
Hence, 

II. Ratios are compared by comparing the fractions that 
represent them. 

If to each term of the ratio 5 : 4 we add 16, the new ratio, 21 : 20, 
is less than the ratio 5 : 4, because \ is greater than f ^. If to each 
tenn of the ratio 4 : 5 we add 16, the new ratio, 20 : 21, is greater 
than the ratio 4 : 5. Hence, 

III. A ratio of greater iiuquality is diminished, and a 
ratio of less inequality is increased, by adding the same 
number to both its terms. 

If from each term of the ratio 32 : 30 we subtract 24, the new 
ratio, 8 : 6, is greater than the ratio 32 : 30. If from each term of 
the ratio 28 : 30 we subtract 16, the new ratio, 13 : 15, is less than 
the ratio 28 : 30. Hence, 

IV. A ratio of greater inequality is increased, and a 
ratio of less inequality is diminished, by taking the same 
number from both terms. 



390 ELEMENTS OF ALGEBRA. 



a c e g 
Suppose ^ = ^=^=^ = r. 

Simplify, br — a, dr = c, fr — e^ hr = g. 
Add the corresponding members and factor the result, 
{b-\-d+f+h)r = a + c + e-\-g. 

Therefore, ^ = 6 + ^ t/t I = ^ = "^ =}" f' H"^'"» 

V. In a series of equal ratios, the sum of the antece- 
dents divided hy the sum of the consequents is equal to any 
antecedent divided by its consequent. 

Notes : 1. The sign : , is an exact equivalent for the sign of division ; and is 
a modification of -r . 

2. A Duplicate Batio is the ratio of the squares ; a Triplicate, of the cubes ; 
a Subduplicate, of the square roots ; a Subtriplicate, of the cube roots of two 
numbers. Thus, a^ : b^ ; a^ : b^ ; Va : Vb; fa : Vb are respectively fhe 
duplicate, triplicate, subduplicate, and subtriplicate ratios of a to b. 

Example 1. Find the ratio compounded of the duplicate ratio of 
2 a a2 _ 
-T- : T2 V 6, and the ratio 3ax : 2by. 

Process, ihe duplicate ratio oi -7- : Tg yB is -r^- : -r^ • 

, . 4a2 6a< I2a^x ]2a*by 
Ihe compound ratio -vg- : -p- ? ^ax : 2by,is — p— : 74 — • 

I2a^x \2a^by ISa^x 12 a^hy bx 

But — To — : TT— ^ = — To i u — = — = bx : ay. 

b^ 6* b^ b^ ay ^ 

Example 2 If 15 (2 2-2 — y'^) — ^ xy, find the ratio x : y. 
Process. From the given equation, a:^ — jV a: ?/ = ^ 2/^. 
Complete the square and solve for x, a; = ^ ?/, or — f i/* 

X 

Therefore, - = a or - 1 . 

Exercise 142. 

Find the ratio compounded of : 

1. The ratio 2 a : 3 &, and the duplicate ratio oi^b'^'.a'h. 

2. The subduplicate ratio of 64 : 9, and the ratio 27 : 56. 



RATIO. 391 

3. The duplicate ratio of 4 : 15, and the triplicate ratio 
of 5 : 2. 

4. 1 — 3^ : 1 + y, X — X i/ : I + s^, and 1 : x — x^. 
a-\-h g'+fe^ {a^-l^f a2_9^^2Q , a8-13a + 42 

Simplify each of the ratios : 

6. 5ax:4:x; li5xy:20a^; 2x^y:\x^. 

7. iaxy.z^ay^; -^ ^ :a^na^. 

Arrange the following ratios in order of magnitude : 

8. 5 : 6, 7 : 8, 41 : 48, and 31 : 36. 

9. a ^ b : a -\- bf and a^ — I? : a^ + b^, when a > 6. 

10. For what value of x will the ratio 15 + a::17 + ic 
be equal to the ratio 1:2? 

11. Find X \ yy if a:^ + 6 ?/2 = 5 a; y. 

12. Find the ratio of x to y, if the ratio 4a; + 5y : 3a;— y 
is equal to 2. 

13. What number must be added to each term of the 
ratio a : h, that it may become equal to the ratio m : nl 

14. What number must be subtracted from the conse- 
quent of the ratio a : b, that it may become equal to the 
ratio m : 7l1 

15. A certain ratio will be equal to 2 : 3, if 2 be added 
to each of its terms; and it will be equal to 1 : 2, if 1 be 
subtracted from each of its terms. Find the ratio. 



392 ELEMENTS OP ALGEBRA. 

16. If a : 6 be in the duplicate ratio oi a -\- x : h + x, 
find X. 

17. Show that a duplicate ratio is greater or less than 
its simple ratio, according as it is a ratio of greater or less 
inequality. 

PROPORTION. 

164. A Proportion is an equality of ratios. Four num- 
bers are in proportion, when the first divided by the second 
is equal to the third divided by the fourth. 

a c 
Thus, if T = -7 , then a, h, c, d, are called proportionals, or are said 

to be in proportion, and they may be written in either of the forms : 

a : b :: c : d, 
read, "a is to & as c is to d! ;" 
or a : b = c : d, 

read, "the rat'o of a to & is equal to the ratio of c to d;" 

a c 
^^ b=d^ 

read, "a divided by b equals c divided by d." 

The Terms of a proportion are the four numbers com- 
pared. The first and third terms are called the antecedents, 
the second and fourth terms, the consequents; the first and 
fourth terms are called the extremes, the second and third 
terms, the means. 

Thus, in the above proportion, a and c are the antecedents, b and 
d the consequents, a and d the extremes, b and c the means. 

Note 1. The algebraic test of a proportion is that the two fractions which 
represent the ratios shall he equal. 





PROPORTION. 


Let 


a:b :: c :d. 


By definition, 


a c 
~b^d' 


Free from fractions, 


ad = bc. Hence, 



393 



I. In any proportion the product of the extremes is equal 
to the product of the means. 

Note 2. If any three terms in a proportion are given, the fourth may be 
found from the relation that the product of the extremes is equal to the 
product of the means. 

Let ad = be. \ 

a c 
Divide by 6 rf, b~d' 

By definition, a:b::c:d. Hence, 

IL If tJu product of tivo numbers is equal to the pro- 
duct of two others, either two may he made tlie extremes 
of a proportion and the other two the 7neans. 

A Mean Proportional is a number used for both means 
of a proportion ; as, h, in the proportion a :b ::h : c. 

A Third Proportional is the fourth term of a proportion 
in which the means are equal; as, c, in the proportion 
a : h :: h : c 

Ia'I a : b :: b : c. 

Therefore!., b^ = ac. 

Extract the square root, b = \/a c. Hence, 

IIL A mean proportional between two numbers is eqv^ 
to the square root of their product. 

Let a :b :: c : d. 

Therefore 1., ail — he, 

a b 
Divide by c rf, ~ ~ d' 

By definition, a. cy.b . d. Hence, 



394 ELEMENTS OF ALGEBRA. 

IV. If four numhers are in proportion, they will he in 
proportion hy alternation; that is, the first will he to (he 
third, as the second is to the fourth. 

Let a :b :: c : d. 

Then I., bc = ad. 

Divide by a c, -::=_. 

•^ a c 

By definition, b : a :: d : c. Hence, 

V. If four numhers are in proportion, they will he in 
proportion hy inversion; that is, the second will he to the 
first as the fourth is to the third. 



Let a:b::c:d. 




By definition, h~ d' 




a c 
Add 1 to each member, t+ 1 = ^ + 1, 




a + b c-^d 
b ~ d 
Therefore, a + b : b :: c + d : d. 


Hence, 



VI. If four numhers are in proportion, they will he in 
proportion hy composition; that is, the sum of the first 
two will he to the second as the sum of the last two is to 
the fourth. 

Let a : b :: c : d. 

. a c 

By definition, J ~ d' 

Subtract 1 from each member, 

a c 

a~b c— d 

«^ -~r = ~d"- 

Therefore, a — b:b::c-d:d. Hence 



PROPORTION. 395 

VII. Jf four niimhers are in propoi'tion, they will he 
in proportion hy division ; that is, the difference of the first 
two will be to the second as the difference of the last two 
is to the fourth. 

Let a :l ::c '.d. 

a 4-h c + d 
Then VI., =-| ^ • 

also VII., 



c 
a — b c — d 



Divide, 



b c 

a+b c+d 



a — b c — d 
By definition, a + b : a — b :: c + d : c — d. Hence, 

VIII. If four numbers are in proportion, they will he 
in proportion hy composition ajid division ; tliat is, the sum 
of the first two will he to their differeiue as the sum of 
the last two is to their difference. 

Let a :b'.:c :d, 

e:f::g:h, 

k : I :: m : n. 
T»,/... ^ c e g k m 
By definition, l = d'f=h'l=n' 

Multiply the corresponding members of the equations together, 

aek c gm 
Ff'l'^dir^' 
By definition, a e k : b f I :: c g m : d h n. Hence, 

IX. The products of the corresponding terms of two or 
more proportions are in proportion. 

Let a:b :: c : d. 

d c 
By definition, h~ d' 

Raise each member to the nth power, fcS ~ ^ * 



396 ELEMENTS OP ALGEBRA. 

Therelore, a" : 6" : : c" : d\ 

1 1 

Extract the nth root of each member, — = -^ • 

Therefore, a" : 6^ :: c" : cT. Hence, 

X. /?i a^iy proportion like 'powers or like roots of the 
terms are in proportion, 

A Continued Proportion is a series of equal ratios ; 

As, 8:4::12:6::10:5 ::16:8; a : b :: c : d :: e :f::g:h, read 
"a is to 6 as c is to d a.s e is to/ as g is to A." 

Kote 3. Four numbers are said to foiin a continued proportion when each 
consequent is the antecedent of the next ratio ; as, a : b :: b : c :: c : d. 

Let a :b :: c : d :: e :/ :: g :h. 

a c e g 
By definition, ^ = ^^ = ^=: ^^ • 

a -^c -^ e + q a c e g 
ByV., (Art. 163), l^d-v f+h^ h^ d = r'h 

Therefore, a-\-c-\-€ + g'.h + d +f -hh i: a : b. Hence, 

XI. In a continued proportion the sum of the ante- 
cedents is to the sum of the consequents as any antecedent 
is to its consequent. 

^^ a2+62 ab + bc 

Example 1. If - .— r i> ^ = ~"a2~i — 2 » prove that 6 is a mean 
ab + be 6^ 4- c^ ' '■ 

proportional between a and c. 

Proof. Free the given equation from fractions, transpose and 
factor, (b^ — acy — 0, or b^ = ac. 

Therefore II., a : h :: b : c. 

Example 2. If a . b :: c : d, prove that m a'^ +^p b'^+ nab : mc^ 
-i-pd^ + ncd :: b^ : d^. 



iVI., 






a b 

c~ d 
a« ab 
7^=7d' 
a* b» 


ab 


nab 




a^ n (t li 


cd- 


ncd' 


or 


c2 ~ 'nc il 


b^ 


pb^ 




a^ pb^ 


d^' 


~ pd^' 


or 


c^~ pd^ 



PROPORTION. 397 

a b ^1 V 
Proof. From the giveu proportion VI., c~ 7l ^ 

a 
Multiply by - , 

Square Loth members of (1), 

By I. (Art. 163), 

Also I. (Art. 163), 

Also I. (Art. 163), r» = ^a- 

m a* pb^ _ n ab _ b'^ a' 
Hence, ^^ =^ = ^^-^ = ^ = -^. 

By V. (Art. 163), ^...^'pd^^ncd = d^^' 

Therefore XL, ma^ + pb^ -^-na b :mc^ + pd^+ ncd :: b^ : d^. 

Example 3. Find x when ^m-\-x-\- ^m-x -. ^m-^x-^m—x 
::n ; 1. 

Procefw. By VIII., 2 ^m-\-x : 2 ^m-x :: n+l : n-1, 

or I. (Art. 163), /^/^Ta: : ^m=^ :: n+l : n-1, 

By X., m + x : m -x :: (n+l)» : (n-l)«. 

By I., {m + x)in- 1)» = (m-x)(n-M)«. 

Simplify, transpose, and factor, 2 n {n^ + 3) x = 2 m (3 nH 1). 

_m(3n« + l) 
Therefore, x - ^(^8^.3) • 

Exercise 143. 

J( ad = bc, prove that : 

1. d :b :: c : a; d : c ::b : a; h : a :: d : c. 

2. hidr.a'.e; c:a::d:h; c:d::a:h. 



398 ELEMENTS OF ALGEBRA. 

Find a mean proportional between : 

3. 2 and 8; 3 and 1 J ; Handf; 8 and 18; a^bsmdah^. 

4. (a + hf and (a - hf ; 360 a* and 250 d^ h\ 
Find a third proportional to : 

5. I and f ; I and | ; .2 and .4 ; 2 and 3 ; f and f 

6. 1 and VI; (a - hf and a^ - 62; ? + ?^ and - • 
Find a fourth proportional to : 

7. 2, 5, and 6 ; 4, |, and | ; f , f , and f ; a, ak, and 6. 

8. a3, ah, and 5a22,. ___, _______ and ^g--^ 

If a : & :: c : 6?, prove that: 
9. a + h : a :\ c -\- d : c\ a — h \ a w c — d : c. 

10. ac :b d :: c^ : d!^; ab : cd :: a^ : c^. 

11. 2a + 3c:3a + 2c::26+3f^:3&+2c?. 

12. 3 a - 5 6 : 3 c — 5 c? :: 5 a + 3 6 : 5 c + 3 c^. 

^oo A7 o ^7^^ + ^ r^, — 6 a b 

13. fa:|?.::|.:H; -^ = __^ . _ = ^ . 

U. 3 a + 2b : S a - 2b :: 3 c -{- 2d : 3 c - 2 d. 

15. la + 77ib : pa + qb :: I c + md : p c + qd.' 

16. ft3 . 2,3 .. ^3 . ^3. ^2 . ^2 .. ^2 _ ^2 . ^2 _ ^2 

17. a2 + c2 : a& + C6^ :: a6 + C6? : b^ + ^2. 

18. V^r:r^:V6::VM^: V^; l = \^P^' 



PROPORTION. 399 

If 6 is a mean proportioual between a and c, prove that; 

If « : 6 :: c : rf :: 6 :/, prove that : 

on J. . i 7.i^_L/ O'+'^c-^Se 2a4-3c+4g 

20. a:fe::a + . + .:^ + ^+/; ^^2rf^3/ = 26+3rf+4/ ' 

21. rf is a third proportional to a and 6, and c is a third 
proportional to b and a, find a and 6 in terras of d and c. 

li m + n : m — n :: X -\- y : X — y, prove that : 

22. ^ •\' w? \ ^ — iv? \\ y^ ■\- n^ \ y^ ^ n^. 
Solve the following proportions : 



23. 




24. 


a:3 ~ yS : (a; - y)8 :: 19 : 1 and a; : 6 :: 4 : y. 


25. 


If = = , show that a + b-{-c = 0. 



X — y y — z z — X 

26. A and B engage in biisine^Jj with different sums. 
A gains SloOO, B loses $500, after which A's money is to 
B's as 3 to 2 ; but had A lost $500 and B gained $1000, 
then A's money would have been to B's as 5 to 9. Find 
each man's investment. 

27. Show that the geometrical mean is a mean propor- 
tional between the arithmetical and harmonical means 
between the two numbers a and b, 

28. When «, &, c, are in harmonical progression, show 
that a:c::a — 5:& — c. Hence, of three consecutive 
terms of a harmonical series, the first is to the third as tht 
first minus the second is to the second minm the thirds 



400 ELEMENTS OF ALGEBRA. 

29. Find the ratio compounded of the ratio 3 « : 4 6, 
and the subduplicate ratio of 16 &* : 9 a* 

30. If - = 3i, find the value of ^ ~ '^ ^ - 



y ' 2x- by 

31. If 6 : a : : 2 : 5, find the value of 2a-Zh:Zh—a. 

32. If T = T, and - = ^, find the value of ^7 ^^ • 

6 4 2/ ' 4:hy — lax 

33. If 7 m — 4 71 : 3 wt + ii : : 5 : 13, find the ratio 
m : n. 

34 If s s— = -TT , find the ratio w : n. 

m2 + 7l2 41 

35. If 2 a: : 3 y be in the duplicate ratio of 2 a? — m : 3 ?/ 
— m, find the value of m. 



a c m 



36. If - = - = -, prove that each of these ratios is 
equal to ^WHH^L 



4 m^c 



37. If 2a + 3Z» : 2,a - 36 :: 2^2+ 37i2 : 2^2- 3ii2, 
show that a has to h the duplicate ratio that m has to 7i. 

38. A railway passenger observes that a train passes 
him, moving in the opposite direction, in 30 seconds ; but 
moving in the same direction with him, it passes him in 
90 seconds. Compare the rates of the two trains. 

Solve the following proportions : 

39. Vx + Vh : Vx - Vh '.: a :h) 2"^' : 22- ; : 8 : 1. 



(x + y-.x 



: X — y '.'. m + n \ m — n* 
40. ^ „ o 2 2 . 2 1 



APPENDIX, 



COMPUTATION OF LOGARITHMS. 

Since the logarithms of all composite numbers are found by add- 
ing the logarithms of their factors (Art. 122), it is only necessary to 
compute the logarithms of prime numbei-s. 

The following method for computing logarithms is the one that 
was used when our tables were first made, although it is not the most 
expeditious method now known. 

Example 1. Find the logarithm of 5. 
Since 10«> = 1, 

and 101 = 10 (1) 

and as 5 lies between 1 and 10, its logarithm must lie between and 1. 

Extract the square root of (1), 10-6 = 3. 162277+ (2) 

As 5 lies between 10 and 3.1622774- its logarithm lies between 
1 and .5. 

Multiply (2) and (1) together, I0i» = 31.62277 f. 

Take the square root, 10 '» = 5.6234134 (3) 

5 lies between 3.162277+ and 5 623413+ , and its logarithm must 
lie between .5 and .75. 

Multiply (2) and (3) together, lO^-^ = 17.7827895914+. 

Take the square root, 10«26 = 4.216964+ (4) 

Since 5 lies between 5.623413+ and 4.2 16964+ , its logarithm 
must lie between .75 and .625. 

Multiply (3) and (4) together, take the square root of the result, 
and we have 1()«876 _ 4.869674+. Continuing thi; process to 22 
operations, we have, 10«8»7(h- = 5.0000(K)+. 

Therefore, log 5.000000+ = .698970+. 



402 ELEMENTS OF ALGEBRA. 

Example 2. Find the logarithm of 2. 

log 2 = log i^ = log 10 - log 5 = 1 - .698970 = .301030. 
Examples. Find the logarithm of 11. 

101 = 10 (1) 

• 108 = 1000 (2) 

Extract the square root of (2), lO^-^ = 31.62277+ (3) 

Multiply (3) and (1) together, lO^-s = 316.2277+. 

Take the square root, lO^-^s = 17.78278+ (4) 

Multiply (4) and (1) together, 102-26 - 177.8278+. 

Take the square root, 10i-i25 = 13.33521+ (5) 

Multiply (5) and (1) together, IO2125 _ 133.352I+. 

Take the square root, 10i-0625 = 11.54782- (6) 

Multiply (6) and (1) together, 102-0625 _ 115.4782+. 
Take the square root, 10i-03i25 _ io.74607+ (7) 

Multiply (7) and (6) together, 102-09375 ^ 1 24.09368+. 
Take the square root, ioi.o46875 ^ 1 1.13973+ (8) 

Multiply (8) and (7) together, 102-078125 ^ 1 19.70845+. 
Take the square root, 10i-o390626 _ 10.94113+. 

Therefore, log 10.94113+ = 1.0390625. 

Continuing the process, the logarithm of 11 maybe found with 
sufficient accuracy. 

Example 4. Find the logarithm of 3. 

Take 10^ = 1 and lO-^ = 3.162277+, and proceed as before to 14 
operations, and we have log 3.0000+ = .47712+. 

A table of logarithms to four decimal places will serve for many 
practical purposes. In the tables most generally used by computers 
they are given to six places of decimals. Seven to ten place loga- 
rithms are necessary for more accurate astronomical and mathemati- 
cal calculations. 



ANSWERS 



TO THE 



ELEMENTS OF ALGEBRA 



BY 
GEORGE LILLEY, Ph.D., LL.D. 

EX-PRESIDENT SOUTH DAKOTA AGRICULTURAL COLLEGE 



'TEACHERS' EDITION 




SILVER, BURDETT & COMPANY 

New Yobk . . . BOSTON . . . Chicago 

1894 



Copyright, 1893, 
By Silver, Burdett and Company. 



John Wilson and Son, Cambridge, U. S. A. 



ANSWERS 



TO THE 



ELEMENTS OF ALGEBRA 



Exercise 1. 

1. a plus 100 ; a plus 10, minus 2 ; etc. 

4. q plus t, plus 8 multiplied by m ; etc. 

6. m-\-n-\-r~t] etc. 

7. m -^ n -\- b\ m + n — b. 

10. X — m ^ n marbles. 

11. 7i + a; -f y 4- 6 4- w. 

12. a-rrti-\-n — k — X — 1/. 

Exercise 2. 

2. A; times ^, plus m times A; divided by c times w, plus a 
divided by 6; or, to the product of k and /, add the quotient 
obtained by dividing the product of m and k by the product 
of c and n, and to this result, add the quotient obtained by 
dividing ahy b; etc. 

Exercise 3. 

1. xyz; 5mn; Sxy-j 15abmn. 

^ ab ab 25 mn r , 

2. — j-T, T. 3. ■ , etc. 

a -\- b a ^ b m -{- n 



ANSWERS TO THE 



a mn 
4. 10. ^. -^. 20 amn. 
o o 

6. 18 mi'. 18 m 71 + 18 m r. 

mn 
8. aoc -\- mnr, 13. 9. . 

G 

11. 25. ^. 14. 5. ^»' + ^^' 

m r 

dt dt dt 

15. It. \- bt. r . 

n n n 



Exercise 4. 

1. m fifth power; 3, m fifth power, x second power; 
etc. ; a, h second power, plus h ; etc. 

3. 10, ah tenth power; m third power, n third power, 
m n third power, etc. ; the third power of m second power 
minus 3 n. 

4. Etc.; 3, a second power, b, times the third power 
of a minus b second power ; a second power plus b second 
power, times the second power of a third power minus b 
third power. 

6. 10 m, plus n fourth power, times the fourth power 
of 10, n second power, minus m fifth power is less than 
15 a times the second power of x, minus y second power, 
times the third power of x plus y ; etc. 

8. m + n. 2x. (a + by. (x — y)\ 5 (x — yf. 

9. (x' + yy. (icH- 7/2)2. m^x^y^. ^xy^. {x'' + y"") {x^ - y% 

11. 1 m'^n^—2n'm^+^a^h'^+^a''h^ + 5a\ '/a-\-b^ 
m — n, .'. (a + by = (m — ny. 

12. .'.X = m^,'.'x + 3m^ = 2x + 2m\ a + a + a 

to ?i — 2 terms = (n — 2) a; etc. 



ELEMENTS OP ALGEBRA. 



Exercise 5. 



1. 3; 448; 60; 180; 64; 9375; 390,625; 1792; etc. 

2. 144; 60; 64; 250; 4; 40; etc. 

3. T^; 3; 7; 75; 32,400; 288; etc. 

4. 6; 60; 2; 3; 5^; 72; 0. 

5. 9; 160; 2048; 81; 8| ; 13^; ^; H- 

8. 20; 11. 12. 2; 0. 16. 4|§f ; 2^. 

9. 0; 12. 13. 249; 134. 17. 1; ||. 

10. 14; 6. 14. 22; ^. 18. 5; 58^. 

11. 18; -14; 2. 15. ^; 36. 19. i; I4. 

Exercise 6. 

1. 11, negative. 11, positive, etc. 

2. is 6 units greater than — 6 ; etc. 

3. 6, 3, 9, 12, 11. 10. b-a. 

8. 2 times the expression in brackets, 3 i in the posi- 
tive series plus 5 a in the negative series, and from this 
result subtract 6 times the expression in brackets, a in the 
negative series plus b in the positive series ; etc. 

10. Value, 26. 

11. Value, 1 + (- a:^^) + (+ a;«) + (-x), 

12. Value, -2. 19. 80; -624^^. 26. 56; -15. 

13. Value, 0. 20. 3; 15^^; 1^. 27. 2; 15. 

14. Etc. 14§^§. 21. 9; 2; 15. 28. 102. 

15. Etc. 17. 22. 127; 21; 6; 1. 29. 52; -18^^. 

16. Etc. 6. 23. 3 ; 6. 30. — 15 ; 12. 

17. Etc. ^. 24. — 4 ; 6. 31. 2 a + b. 

18. Etc. -2r»^. 25. 16; 55. 



ANSWERS TO THE 

32. 5« + (6 - 1). 33. h + ^±_^. 

34. x-\-x-{-x-{-....toa terms, or ax. 

35. n, n -{- 1, n -\- 2. 

36. m, m — 1, m — 2^ w — 3, m — 4. 

37. (a — 1) m ; (m + w) m. 

38. X — S, X — 2, X — if X, X -\- 1, X + 2, X -\- 3. 

39. a- -2/". ^ , a; 

40. 2x + '^ + abc, 41. ^. 

b y — c 

42. a" + - — 5 ( ~\ >h — x. 

43. (cc^ — ^»« 4- 2/'") ^ ^" < g'^". 

a** « — ^> 

44. x"" — 



b"" a' + b^ 

(A 

45. — ^ — .X ?/ + cc + X + cc + • . . • to w terms — a"*. 

46. x"* H ■=x — y-\- (4a+^» — m) + — . 

47. 5^8 — 3 6i^Z>8 + 2^»2. 49. %x^y'^ — a^b\ 

48. 3^2- 2cc«?/ + ^3. 50. + 6a(ir 4- 2/-^). 

a: — y 



Exercise 7. 

1. (+15.1 a). 5. (-5£c2). 9. (+3.8a-^c(/). 

2. (+15 ax). 6. (+/2^). ' 10. -7.71 (^' + c). 

3. (+41c). 7. (-21a8). 11. + 5.81 (x - ?/)8. 

4. (+ 2 a ^ c). 8. (- 3.5 a" U^). 12. - 3/^ (|) . 



ELEMENTS OF ALGEBRA. 



Exercise 8. 



1. (-6.69 a;) + (+3.5^). 4. (+2^ x') -\- {+ i ab). 

2. (+Ha) + (-5'5«^)- 5. 14.9 a- 2.67 (x-y). 



3. 



(+8a'a:) + (+7c«x'). ^ 6.1Q-2.3Q. 



Exercise 9. 

1. 5 y. 2. 4 // — 5 a. 3. x. ^. a + b -\- 2 x -\- t/. 

5. —33 ax — 4:bd -\- 5ni7i. 

6. -2a + ^ + c + </ + ^'i + a^. 

7. — 8 a i c -j- 53 a 6 m — 20 c m. 8. 4 w. 
9. 3 a -h 3 /> + 3 c + 3 rf. 

10. 4.25 a^ 4- 3.3 f + 2 6. 11. 2{^^a+l^^ab. 

12. iVo ^'^■^ + -'^l ''^ — 2t\ w n — 2^. 

13. — ac+1.92frf. 14. — (1*^.5 a^b+ .25 a b^-{-b\ 
15. 7.8 (m — ny — 6.03 (« + y)^- 16. a^ 6» + x^y, 

17. 7tJ5 a^ + 5|i « a^« + T^^ //« + 3^ x^ U- 

18. I «• 4- J\ Z^'' - 1.3 c» - «2^ + m a^c-.2ac^-^l}fab^ 

+ ljgga6c + 2*3c-1.3^c'». 

19. 21.33 a^ + 8.37 ar^ - 5 x -f 8.5. 

20. 2.6 a* b^ r» + 2.8 «« 6» c^ + 3.91 a^ 6» c*. 

21. 6.09 a* + h f' + 4.97g 6 c + 3.5 c"* — 3.03. 

22. \aX"l.Sab + 5.2x + ^xy. 

23. 12.91 a + 4.1 y — 6.82 z. 

24. 3 o 6 + 10 J- y 4- 9 a- // — .T« y + 22. 

25. - J(m-3a-)". 

26. 5 a*. 29. <?» + ^.« H- c« — 3 a 6 c. 

27. 10a« + 86«+12a + 12. 30. 0. 

28. x* — y*. 31. 4 a:'" + 2 a" — a". 



ANSWERS TO THE 

Exercise 10. 

1. lOaHc; -SSab^xi/, 7. ^ (x + y). 

2. 2x'y^; 0. a -3ax\ 

3. —daxy, xy^ + bc. 9. 26axy\ 

4. —3.75x\ 10. %x^y. 

5. -2.72 a bc^ 11. 8.9i(a + ^)2. 

6. —13.1 VI np^^ 12. 4TV5ra^". 

Exercise 11. 

1. —2x + 4:y — 3z. 5. ~-2x + y — :^z. 

2. -2x-y-4.z. 6. -^.T + |2/ + i 

3. —2a — 10c. 7. 2a^ + 3aH — 3hc—10, 
^. Ix+^y + l^z. 8. 2x^y + 2abx + l. 

9. abcxy — 3aby-}-5bx — 3. 

10. 2^372/ — 5ac?/ — 2a^c + a + l. 

11. .6 ic* - 1.8 x^ + 7.03 cc - 9. 

12. —x^ — 1.2x* — 2x^ — 2x-\- 2. 

13. ^^ 77Z8 — I y«, 7^2 _ J s ^2 24. 7^8 — «. 

15. ^^m^--J^y + 2lin-^x. 

16. 2.5«2^,c — l^a-?/"- 4t^c. 17. 0. 

18. 2 a;*" + 4 ;?;" y"* + y"^. 

19. 1.8 ^/"^ x^ - 1.7 a ^»3 .X + .37 b"" c x"" + .7. 

20. a^ _,_ 3 53 _ g ^4^ 21. 3.55 ic^ + 33. 2y^x + S.5x z\ 
22. ic'» + 2y"'. 23. | iy _ 1.9§ 2/^ + 1-7 ?/« + 3. 



3. 2-2a^-2a-2. 

4. 2 a ^ — ^2 _ ^2^ 

^2 n yin rjf.Zn 2 ^*" 







Exercise 


12. 


1. 


— 4 m^ — 2 w - 


- 5 7z2 + 6. 




2. 


4.x^-Uy. 






5. 


— 2 m. 


6. 4?/ 


w-l 



ELEMENTS OF ALGEBRA. 9 

7. -xhyi-ei/^ + Sz' -^1. 

9. - ^ ari + 2.9 a;* yi - 33 yl 

10. 3.9 cyi + 1.6i ax + .31 ^» - 1.2 7m - 5^%. 

11. 5.5 6" — 3.25 «"• — 4 w. 12. y- — x^ — 1.2 i". 

13. ^ c' — l.Ta'" — o^b"* — 2 d". 

14. c» - i (a» - by - 12 (x» + y2^«. 

15. 2 a^ 21. 6 rt- — 2 a ^. 

16. 3 a;«* + 2 x^* — 6 a;". 22. aJ ^i. 

17. 2^2;— I a^- 2. 23. 4y a^- ^a» — 4.5a + 3f. 

18. \ //i« +1. 2^. llbc -\-l mn — ^xy. 

19. 7 J/'' + 18y — 4. 25. 4 x*-^ — 2x'''-^. 

20. 0. 26. x"— 4. 

27. 64 (x 4- y)* + .3 a- + x"* - 3^ a-x" - C + 3. 

Exercise 13. 

1. 7x8; 15a«x«; 2a'*^*x«. 

2. ix"y»«m««"; 2an»c«<;»w»w». 

3. a6»c»y"^*; ?a"x»y"«. 

4. 30«"x*y"; ix'" + *y"+^ 

5. 2a-^'/>^V^*»x^<>y"; Ti5a'Z»"r"j'" + »y"+». 

6. vSa"+>6'' + ''x^+V"^*; ^'/A 

7. a'" + -6»+»'; 5.7 x*y". 8. a»6»x«y«; a^-ft^". 

10. 2a»x»y". 15. .765a«mM+'x— "-'1. 

11. 3a*'+''6" ♦c'+'"rf*. 16. l|a'"+*'-**^-"//"+*«'". 

12. 2a* + »c*+-+*rf«x'"+". 17. l\(a-\-hy\ 

13. 2 a*l m^ n** x*» /. 18. 3 (c + ^)><> (a + ^)". 

14. 5 aHi a;*A yi. 19. ^ (« -|- ^,)-+ *» (x — y)'"+«. 



10 ANSWERS TO THE 



Exercise 14. 



1. —2S10anG', a^x^'f. 6. -3a-H-^c^^d. 

2. w'hx^y'; —ha^'^hU^d}. 7. an-^(Px\ 

3. 4:aHcxy) .Sacm^x^T/^ 8. — .3 a^^ x^ yi -^ aH x^. 

4. ^a^bcdx^yz; a^^+H'+^x^"" y\ 

5. — 3 a^b^x^yzvw, a'«+" + i^>" + 'a:"* + ^ ?/"+\ 

9. — m^ n x^t yH '^ — 2.1 ai bl 11. A X 3" mx^ y'^^'p q"". 

10. a^'^ ''■ a aa--- -to 10 ni3iGtovs. 12. _ 2^« + »"3^°« + ^" + i. 

Exercise 15. 

1. abU^+aHc^-aH'c, ^ aH'* - :^ a^^^ c^ - ^ aH^^ e^ 

2. 5 a^ Z*» c^o — a^ b'' c"" — 2a^b^ &'' ; .12 ic^ y^ - .1 £c^ / 

3. 3m^ n — 1\ w? ii^ + 3 m ?i^ ; 9?- y — xy'^ —l\x^ y^. 

5. 9 a«Z»ar3— |«H^aji3^ .3a^>«ar«; p^ x'^'^r — p qrx'^-^'' 

— pr^ ajtn 

6. 3«'"^>2_2«^« + 4 rt»» + l^n + 2. .Ga^'^Z;^^ - 2 6/"'-^^>2p 

8. a^i ^^s - a^^ ^»^^ - a^ b^^ + a^b; x^ y-h - x^ y\ -\- x^ y 

— .Bx^yl 

9.-2 .T.3 ?/^ + 8 a^2y6 _ 8 _^ yv . ^„2 ,^o _ 2 ^^p ^2? ^ ^o ^^2,^ 

10. ^V «^ ^'^ ^^ + ^7 «* ^ cc' + ^ a^ Z*^ ic^ 

11. — cc§ ?/ + .-^^ ?/^ ; ai ^'«^ — 6?i b'^\ 

12. 45a!i3i7/°i-45£c'«^/3>. 

13. ^V ^'^^'' — :\ ^'''^"■•'^'^' — -2 ^>'3\tV 4- ^ //7-5. 

14. 180 a-Sm/.-am _ 189 ^-4m ^--im _ |4Q ^-2m Jl-2m^ 



ELEMENTS OF ALGEBRA. 11 

Exercise 16. 

1. a*-{-aH^+b*; a* + 4 a^x^ + 16 x*. 

2. x^ -\- i/; ^xij. 

3. 2/' + -'y'~lly'-12y + 27; y* + a y» + « V 

— a t/ z — a*z — a*. 

4. \x*-l,^^x' + ^jf', .64 a» 4- 3.24 a ^2 - 2.7 ^»». 

5. a^-{-2x'-2x'i/ + x^-xi/- 7/ -{•!/', 

l^x*- 1^ ax« + i a^x^ — I a\ 

6. a;« + 8a:y + y — 1; H a:^ + 6 aic* + f a*x« + faa^* 

+ 3a^x^ -{- la^x _ ^ a«x« — ^ a«a- - j^jj a». 

7. a-' - 7 a;« + 21 a;* - 17 a-* - 25 a-« 4- 6 ^2 - 2 a; - 4. 

8. a» + 2^i«4-5a* + 2a24. 1. 

9. x» -h 1.25a;' + .25a;« + Sx*^ + .5.r* + .25 x' + 1.25 a:^ + 1. 

10. 4 + 32 a - 4 a^ -f 25 a» - 6 a* 4- «». 

11. x» — 32 y*. 13. a« — 3 a ^> c + // 4- c». 

12. x^* 4- y^^ \ x^ -\- i/. 14. «» 4- 3 rt ^» c 4- ^' — c*. 
15. a*^»2 - rt^-2 4- ^*<^ 4- ^f/^. 16. — 8 a-2 //. 

17. .3 m« 4- 2.90 « w — 3.01 i m - .3 71-^ 4- 3.01 a 7i - 2.99 b n 

— .01 a^ 4- .1 ^. 

18. a^x^'* + b^x'^'* + 1^ -\- 2a^»a;'" + " + 2ara:" + 26rx"; 

ari — yl 

19. a*" + 2a'" 6- 4- b^"', a*"* — i*-; a:l 4- &^*+ J5a-*4- ^'. 

20. 6a;'" — 4a;;/'-*~9ar— V* + 6y; a^a-m+a _^ ^^^,3.n+2 

+ a^6x* — a^ar'"+' — i'^x-^'^-ff^^'^a:* — ax"" — ^x" 

— a Ax. 

21. 3a*'"'x4-3a2 + -y + f(2''^'" — Sri^'^+^a; — 3a'-*-''?/ — a'" 

4-3a«-x2 4-3a«xy 4- «*"^; x5 — x=//-? — xiy + y*. 

22. .04a — .09 61; x — y. 24. l + x + x^ + x^ + x^ — x^ 

23. X* — i/-\ 25. X* — 5 a* x* 4- 4 a*. 



12 ANSWERS TO THE 

26. 120 ic* - 346 x^ - 205 x^ + 146 a; - 120. 

27. x^ + x^ + 1. 28. x^ + a*x^+a\ 30. a^"*— Z.6«, 
29. 3a^-oa'b-SaH''-j-7 aH^ + 6aH^-2ah^- h\ 

Exercise 17. 

1. a2^2a-15; &2_^^>-30; a;2^7a; + 12; a;2-3a;-4 

a;2 - 5 £c - 14. 

2. ^2— 14ic + 48; a2+4a-45; a2_4^_32. 4a;2-18a;+20 

9ic''^+ 6aj — 35. 

3. :c6 — 7ic^2/^ + 12?/^; a;^ + xy — ^y'^; a^"^ + a"" — 2 

9:i;i«-27a;5 + 20. 

4. 4a*/-8ay-32; 9 a^ a;2_^ 9 a ic - 28 ; a;« - a a^^-12a2 

• x'o _ ^2 ^5 _ 6 ^-2 

5. 4a;2_2(ia;— 2^2; 4x2" + 4 a x*^ — 15 a^; 9^:2 — 60:3/ 

— 2 y^; Ax^ — Amx^ — 24: rn}. 

6. a;2_6^^_|_5^2. ^2^3 ^ ^_4q^,2. ei^ _- 8 a^ £c + 12 a;^ 5 

25x2»-5a2;K^o_i2a^ 

7. 25 a:^« - 25 cc5^2 ^ 6 2/^ 9 a^« + 3 a^ (2 a ft - 4 a ^/) 

-%a'b'; «2" + a" (3 - 6) - 3 Z». 

8. 16^2 + 4a(^>- c) -^»c; 2na' — 10a(b^c) + 4.hc', 

a^y'^ + I aa;^/ + ii^^'-, «^ + i «' — I- 

9. 4ar + 26 a;^ + 12; 4 a + 2 a^ (5 — 3 a a;) — 3a ^»a;; 

.09 a; 2/2" - .3 a;^ ?/" (a; + 2/) + «^ V- 

Exercise 18. 

1. 4a;^-92/^ a:^ - 4 2/^ 25- 9x^ 25 a:'^ - 121. 

2. 4 a;2 - 1 ; 4 a;2 - 25 ; 25 x^y' - 9 ; c^ - a^. 

3. c* — a^\ m^ri^ — 1 ; a^ y^ — }p-\ a^ x^ — 1. 

4. x^ — y^; 1—p^q^; m^ — n^\ a^"* — a^". 

5. 25x2?/-2-16 2/^ 25a!^-9y^ x^-9x\ 



ELEMENTS OF ALGEBRA. 13 

6. ^a'^x' — b'^y^', m-^^ — ii-^^-, 100 a" 2"' _ 169 6"''*. 

7. m + w; 16a — 400a;*; al — b-%, 

8. 121 a:- 900 y; 225 a* *•- 256 a 61. 

9. \aH-^-tb-^x-^; a^-b\ 

10. aH^-l\ 16 a*'" — 256 a*". 

11. 625 a»2 - 1296 b^ ; a" ^^ _ «« 6". 

12. x-\-x-'y^; |ec-'"-^af^"*- 

Exercise 19. 

1. 4 a** — a-" + 6 a" — 9 ; 20 a^ — 3 x2« — a;'- — 6 x^**. 

2. 4 a*' 4- li a" + 3§ a^^ + 6 a' - 15 ; 2 « + ^5 a;5 - | a;4 

3. _ § a» + 6^ at — § a2 — 4^ al + 6 a* — a« + I a-l - a-2 ; 

6 a*"—'*' + 3 ar--^b^f — 2 a^-sp^-sp _ ^p ^-p 

4. 25 a^V* + 20 a;2«2/26 _ 15 a:« y* _ 12 x« y% a^* + a" 

+ a— 4- a-*". 

5. .09a«-.156a*6 + .22a*6«-.488a»6«-1.39a«6*+.3a6* 

+ .rib\ 

6. 1 _2a;J-3a;i + 2aji + 2a;i; a\ — al ^- 4|al-8ia-i 

+ a-i - 2 a-i + i a-» - i a'^ + «-« - 2 a"*. 

7. -2a;t + 2x5-3x-5a;-l + a;-84- 3a;-«; al"+a-i". 

8. x""+' — 2a:*"+'*' — 2ar2''+2 4.2a^'' + ' + a:''^* + 2 3f + '— a^+* 

- 2x* + x"-'; j;2n+2_4^2n _^ 12a:2--i — 9x''*-*. 

9. 2a:»"-*-4a;"'-' + 2.1ar«»-.9ar*''-'+.lx*"+2+.2a;'»+* 

4. 2a;»"-» — Sar*— 2 + x«— » _ ar^-. 

10. 9af+"-' — 34a^+— » 4- 29 ar + - + » — «•"+"+*. 

11. 2a;^+> — 63:*+* 4- 2x«+'^ — 4a:'+* + 3a:*«+» - 9 a:»«+* 

+ 3a:*^+» — ^Q^'^''-^ — 4a:»'^+» + 12x«'+* - 4 a;'«+» 
+ 8a^'+^ 

12. ITar^-^V*"^*— lOx^^+V + l^x^'-^V"^'— ^^"""V*"^* 



14 ANSWERS TO THE 

13. m^ + '-— 3m^+'-i 7i+ m^+'-^Ti^— m'+'-^Ti^ _ 3^p+r+i ^ 
+ 9 mP + '- n^ — 3 mP+'-'^ n^ + 3mP + '■ -'^71^ -{- 7iiP-^'-+'' 71^ 

15. 2 x^y — 18 x-^^^ + 6 a;3y _ 18 a:'^ / + 4 a:?/^ 

16. f — x-^"*; ^^x'-y-'^b _^^^-'zay2i,^ 

17. x^"— 2/2m. ^_^^_20; 49x2 — 9^-2, 

18. 16 x2 — 8 x-0 - 15 a:-2 ; ^cH-^ — ^\ ^» Z/"* ; a^n _^ ^14 ^n 

+ 49-9a-2«. 

Exercise 20. 

1. 16aH^', 21a>^m', 32x^y^', .00032 a^o^^i^c'-O; .01a2»^,2«. 

2. 49a«^^ 121 a2&4c«<^^; - 27 c^ a:« 5/I2 ^i^ . 27a^^^»«y^ 

25 a^^^^^^yo ^20^ 

3. 256 a« 6« ci« ^^^4 ^^32. ^10 ^10 ^10 ^10 ^10. _ ^^9 ^e ^s . ^8 js. 

729a^6^8c^; -^aH\ 

^4nj.3n^«^2n^ 

5. — 8a-^'7»'«"; m"=; ««'; a^b-'^c-^^', m""*'/!-"*'"; -8; a^n^ 1. 

6. 16a*5-«c*?^^; ri^^ + i^smn^ ^»»« + ia-m. ^rri'n^x^^f. 

7. -54am3 7i»a:^2. 81 a^« 5«^ c^* m^^ ; 81 a^^,-*. a«^-« + i. 

-243a^-20c-'#ic*?/§. 
9. _ ^14^21 ^56^28^. _a^rm,/»5 ^-'"'b^'k^*; _8a2i^,i4. x-^y^\ 

10. X^2«^42m. ^21^65. 4" ^7n J7n ^7»« . _^7m»n«. J^Uk ^Ukm^ 

11. m" (a — 3 <^)P" (x — 7/)'" ; (a — 3 c^)i"' (a; — 2/)^" ; 

3" (a — & + c + c?)" (a — cc)""*. 

12. a" ^>" c" (a — &)"*"(cc + y + z'^Y" ; ct"' (x — y + ^)^"' (cc — y"»)8'»*. 



ELEMENTS OF ALGEBRA. 16 



Exercise 21. 



1. a;« + 4 X + 4 ; m^ -\- 10 m + 25 ; n^ + 14 w + 49 ; 

a«-20tt + 100; 4a;* + 12a;y + 9y^ a"" -\- 6ab + 9b^; 

2. J-2+ 10xy4-25y2; 9x«-30a:?/ + 25 2/'^; 4^2 + 4^26 

+ a-62; 25x2-30x^^ + 9x2^2; 25a2^,2c'2— lOaic^ + c*; 
a;iy-i_4icy + 4?/*; a^m ^ Qa'^b-" -{- ^b-'^\ 

3. 4 x^ 4- 12 a* X + 9 a ; x^ f/^ + 2 x^ y -{- x-t -, 9 a-* 

+ 30a»6-» + 25ai»^»-«; l-2x + x2. l_2cy+cV; 
m2-2w + l; a2^,4_2a^-^+l; || a^-- /^ a* + ^i^. 

4. ^a2^-*-f 3a6-»x-» + |6-2x-2; j^*'*^*'- — 2j9*^'"r«4- r^'; 

.00000004x2'" + .000002 x*"//" + .000025 y2». 

5. 3»j/wMtV — 2w»7i'' + 'y/ + Y^^''^^"'; icV + 2y2a;« 

+ //2^2_^3.a-g2 _j_ 2ar2y^ -I- 2xy«2. 4x* + 5x«+l 

+ 12x»-6x; x* + 6x2+ l_4x» — 4x; x*-4x2 
+ 16 + 4 x» - 16 X. 

6. 4a?*H-13x« + 9-4x»-6x; x« + 25x* + 4 - lOx* 

— 4 X* — 20 X ; 16 w* 4- w* ^2 + 7i» 4- 8 m^ n* — Sn* 

— 2m^n^\ x» + 9x2 + 4__6x« + 4x* — 12x. 

7. x2 y2 _|_ 4 ^8 ^ 1 _ 4 ,j jp y _l_ 2 a; y — 4 ?i ; rti^ ■\- v} ■\- j^ 

-{- q^ — 2 m7i — 2mp — 2mq-\- 2np -\- 2nq -\- 2pq] 
a;« + 8x^ + 16x2+ 9-4x'^-14x^-12x; 1 + 3x2 
+ 3x* + x« + 2x + 4x« + 2x»; x2 + 9^/2+ 4^2 + ^2 
+ 6xy + 4ax — 2/>x + 12 ay — 6 bi/ — 4ab. 

8. 4x*+5x*- 17x2 + 9 + 18x»-6x; x2+4y2^9^2 

+ 4n2 — 4xy— 6x2 + ^ n x -\- 12 yz -\- ^ n y — 12 nz\ 
^2" + n2"* + />2» ^ ,^2« ^ 2 ?;^'•7^"• + 2??^''/>" — 2m* q"^ 
+ 2n'-/)» — 2n-^'" — 2/>»^'"; ^a^ + g^ + f — 2a6 

— a + 9 ft. 

9. 4a« + 4 62+TVc2~2ai + iac-Ac; x2" + y2m _,. ^ ^2 

+ i 62 _ 2 X* 2r + <» X- - 2 6x* — ay» + ^ *2r — J\ aft; 
^.r* + 3x2+ |-$x«"-3x; |a:«+}x«+^-2x» 
+ ^x*-§x». 



16 ANSWERS TO THE 

10. 1-f- ia:+^V^'; i^-'-^^' + ^V-i^' + Aa^; ^^a'^ 

4 a;*3 + 25 x + 49 + 20 a;i + 28 ici + 70 o^i 

11. 9x + 4.x^ + ^xi-{-x-i~12x^ + 2a;i — 6a:J — |£ci5 4-4ccs 

5-23 x3i 

Exercise 22. 

1. a'-7 a«^» + 21 a^^2 _ 35 ^4^8 ^ 35^8^4 _ 21 ^2 ^5 ^ 7^^.! 

— ^>'; a^ + 6 a° ic + 15 a^x^ + 20 a'^c^ + 15 a'ic* + 6 aa:^ 
+ a:«; a^-4:a'c+ 6 a^ c'' - 4: a^ c^ + aU^; a^-16a« 
+ 96a« - 256a» + 256; 16 + 32a + 24.a'' + Sa^ + a^ 
a^ - 5 a* + 10 a^ _ 10 ^2 _^ 5 a - 1 ; 1 - 5 a + 10 a^ 
~10a» + 5a^-a^ ; 16 a*- 96 a^i + 216 a^b^ - 216 ab^ 
+ Slb\ 

2. ic2-12ici+ 54ic-108xi+81; a^a:^ -15a4a;« + 90 a^a;' 

- 270 a'x^ + 405 ax^ - 243a;^«; x' - 15 x' + 90ic« 
-270a;2 + 405ic-243; Sa^x^-ma''Px''i/-^54:aH'xy'' 

- 27 ^« 2/«; 16 a* o:^ + 96 a^ ^ a;^ y + 216 a^ V x" y'' 
4- 216ab^xy^ ■\- SI b'y\ 

3. a« 4- 12 a^ + 60 a" + 160 r^^ + 240 a^ + 192 a + 64; 

a« - 12 a^ + 60 a' — 160 a^ + 240 w" — 192 a + 64; 
16 — 3^ a 4_ I ^2 _ ^a^ ^8 _^ ^ ^4 . ^1^ ^4 _ 3 ^3 ^ 

+ V- <*''*' - 54.ab^ + 81 ^4 . _i^ «4 _^ ^ ^3 ^ _j. ^ ^2 ^2 

- 3^ a 6» + ^V ^^ ai« + 10 «» ^ + 45 V ^2 + 120 aH^ 
+ 210 a^ b^ 4- 252 a^ b^ + 210 a* 6« + 120 a^ b' + 45 aH^ 
+ 10aP + b^\ 

4. a + 4 4- a-i — 4a^--2a«+4a-i; 4x^+4^4-8:^2^2. 

l + 3a2_^^4_,_2a4.2a^3; _l_5a2-4a* + 2a 
+ 4a». 

5. 16a8i + 64«63; _7 + 20a-16ct2^4a«; 1 + 2 • 3^ • 5J 

^ 2 . 2i • 3^, 



ELEMENTS OF ALGEBRA, 17 



Exercise 23. 



1. 3a^bj 6ab\ ^abc-^\ f mA. 

2. w«; a^ a-H-^c^-''', a?"; 2^'. 

3. 5al AJa;-^ ^a-i**; 3a••-^mx'-^ 

5. (5c — y)2; (a-c)*; | 6*-i r"' A;«-«. 

6. Oa^^c-^ a'"'-"'; 2'm'w"^ 

Exercise 24. 

1. 2a;; -2a*6*c«; -oa*; 1. 

2. —3a; -^a^i^c*; .Saft-^c"^; -SOa'^ir*. 

» a A 5 m~ * a;* V* n i ^ 2 s 

*• — ^^ «7wy«; — 31 ma: 2/^"-*"'. 

5. -6a:"-'/''; - ii(a-^)c"^ ^a-i^^-i* 

6. 10 a;l y- i (a;i — - y') A ; | w w x A y. 

7. - 4 a* (x - y)^ 2- 1 ; m" n"" (a; - y)^'' (y - «)H. 

8. 6a-^6c; -2a'» + "6- + "c-'. 

9. -2a-Sil. 

k 13. -^a^-^-d^-^-c*-"". 

11. 2a6*c*rf°x-». 14. Ha"'6''-ic-Ix-». 

Exercise 25. 

1. 1 + 3 ay — 4 aV; 3 m^ /i^ - m 7i - 2 + -?- . 

mn 

2. 1 -J«c- aft r2 + aH*c^ 6a;*— V- a: 4- 40. 

3. 4a«+ f a-3 + -; 4a6-6-^-a-^ 

a 

4. 9aftr« — 12aft' — 5c*+ ^aftc. 

2 



18 ANSWERS TO THE 

5. 3mn — rn^n"^ — 3 n -{- 5 th n^ -, x"^ — x^^ ?/'. - 

6. 2 a — 3 ^> + 4 c; — J3O- ^^'^ + 2 n". 

7. _ 3 ^4^7 _ I ^2^-1 ^ 2 a-i; .9 7i^e^ - 1.2 ?iA. 

8. 36f 05 2/" + 10-^-8^; 3mi« + 160m-80 7y^-l7^. 

9. ^m-2 _ 2 a-«>-2 4. 3 a!'-^\ m"-2 — m**"^ + ??*" — w" ' * 

10. — 3 xy""-'' — 2d^x^ + 4.a'^x if''. 

11. _ a»»-i y^ 4. a"* ^> — a'*-^; — f a^ ic'^ + \l a x^. 

12. 6 a — ^ ^ — 6'; xi — 3 ars". 

13. 3 m5 — f ?j3 + ^ r^if ?ii 

14. 4 (x - ^)* - 3 (x - y)2 + 2. 

15. 2«'-i + Sx"'?/" — 18a:'"y-'*^'+\ 

16. (cc — ?/)''-'* — m«-". 

17. (x + yf-' (x — yy-" + (X + yy-'(x — 2/)«-^ 

18. 3 7?i — 2 71 — 4 ; «i — a*^ ^>2 + ai 

Exercise 26. 

1. Tx'' + 5xy + 2y^ 

2. x^ — x"-^ ?/ + £C z/'-^ ; a^ + or ^ — b'^. 

3. vy« _ 2 2/2 + 7/ + 1; 7/5 + 2/* + 2/' + 2/' + .V + 1- 

4. JC+22/-.^; a7 + a6^_^^5^2_|_^4^34^3^44^2^5_^^56 4.J7^ 

5. 2 a; + 3 ^ ; a^ 

6. .25 x2 — 3 X ?/ + 9 7/2. 

7. 27x2 + 12x7/+ 6 7/2-1; x^ — x^ 7/ + xif — y^. 

8. 1 — 7/ — X + XT/; 2 7/3 — 3 7/2 + 2 7/. 

9. x2 — X 7/ + X ;2 + 7/2 + 7/ ;>; + ^2 . ^.3 _|_ ^^ yj 4. /jjj 2/i + 7/i 

10. X 7/ + 7/ ^ — X ^ ; X'^" + X^ 2/^ + aJ5 7/t + X? 7/5 + 7/t. 

11. 4x2 — 6X7/ — 8 7/2. , ■ • 

12. X2 — X7/ + X + 7/2 + 7/+ 1; ^X2-^X + yV- 



ELEMENTS OF ALGEBRA. 19 

13. 6x*y«-4ar^y« + y^ x! + xyi + xiy + yl. 

14. aH — ab^j x» + x-\ 

15. X- + a; 2/ + y* ; « + "i Oi + 0. 

16. a^- ia+ 2; 2*5^*-3«'y+ ixy\ 

17. a — aJ ; x* + x*y + x'^y" + x ly' + y^ 

18. X* + y-^ + ;g'^ + - xy - X « — y « ; x'^ + .75. 

19. x4-2y + z. 22. 2a^ — 3ab + ^b\ 

20. |x*-ia;-§. 23. 6aj-^y-i. 

21. X* — x'' y + X y* — y** ; x'— x* y + x^ y* — x' y* + xy* — y'. 

24. a»-2tt«i'- + 3a*^»^-2a-'^° + ^«; a»+ 2a^i + 2 a6^ + ^»». 

25. 2x** — 4x-y" + 22^"; x^'^-x'-y" ^y'". 

26. X* — y" + z' ; 3' + 2'. 

27. |a^ + ix»y4- ^% x'y^ + ^h^y'i x-i'» + 2y-i". 

28. yx" + zx" + r. 

29. x-i + y-*; x^ — 2Jxy + y'*. 

30. x-J-y-l + sr-i; x» + x''y + xy»+ y'- ^^' 



X — y 

Exercise 27. 

1. 7/1* + //? n -f J/'* ; a* m* -{- a^ bm^n-\- a^ b^ m^ n^ -{■ ab^mn* 

-^ b*n*; m* n* + w^ n^ + m^ n^ ■\- mn -\- \. 

2. 1 + m n X + m' n* X* + 7w* 7i* x* + w* ;i* x* + m* 7i* x* ; 

x«»y« 4- xV« + x«y 2* 4- x\//2* + xV^* + a;«yz» 
4- x*2% 1 + ahX'\-a''b'^2? + a''Vv+ a*6*x* + a'*6«x'» 
+ rrV/«x«. 

3. a«+rt»^ + ft*; x" + x'y' + y"; x" + x^V + x»y< + x«y« 

4. a" + a»i« + a*i"+ aH^.+ i^; x""H- x"- 7/- + .r'*"?/"'- 

+ 2*-x*- + 2*-x-"' + x*-. 



20 ANSWERS TO THE 

5. 16 a^ + 12 a^n^ +9 n^', 4:x^y^*'+ I. 

6. a'^x^P + a*b^x^pf"' + aH^x'^y^"' + b^y^'"', 16x^+2Ax^y^ 

+ 36x'y' + 54.x'2/ + 81^1 

7. x-s + ic-K?/-^ + x-^^y-i + y-i\ xi + ic-^/^ + »^^y + ^V^ 

+ x^iy^ + y}', a^ x?+ ai b-^x^ y^s + a^ b-^xyi+ai bx^yi + b^y^. 

Exercise 28. 

1. 12ba^x^— l^a^mnx'^ + ^5 am^ 71^ x— 21 m^ 71^', x^-x^b^ 

+ x'b^ — b^; x'' — x^ + x^ — x^ + x^— x^ + X — 1. 

2. £c5 - 2/?; 64 a;^ - 160 x^ + 400ic - 1000; x^"" - 7/\ 

- 1^5 2/^ ; a^-aH + a' h' -- aH'' ■\- cv' b^ - a'' b'> + «' ^' 

- a^ ^7 + «, ^8 _ ^9^ 

4. 243 ai<> - 162 a^ b^ + 108 a« b^-12 a^Z/^ 48 a2^»i2_ 32Z;i5; 

ax" — 5'-^ 2/2"*. 

5. a-^ — a-^x-^ + a- 5 ic- « — X- ^ ; a^ x~ ^ — a^ ^s x-%~i 

-\- ab^ X- -^ y-^ — b^ y- 1 

6. icl« — £c"2/T2'" + xi'*?/i'" — x^"2/i'"' + xi"?/3'" — ?/?5"'; 

Exercise 29. 

+ ^-22/-2_.^-iy-3+ y-4. 2o6x'-192x'y + lUx'y' 

-10Sx7f + Sly\ 

2. 64xi«- 96x1^7/2+ 144xi2y4_2i6x« 2/'+ 324x«2/'-486xy^ 

+ 729//2; Slx^^-5ix^7f + 36x'y'-24.x^y'-{-16y\ 

3. Xl2n _^iou ySm _|_ ^8«y6m_ ^6« ^^9m _j_ ^4« ^12m _ ^2n ^15m 

_|_ yl8 m . 7^18 a_jA6a ^^5 n _|_ /^12 « ^^^10 ri _ ;^9 a ^^^15 n ^ ^6 a^,^20 » 

- A;8« ^25" + m»^" ; a^^ - a^ ^' + a^ b^ — aH^ + a^ b^ 

- aH^ + aH^ -aH' + aH^ - ab^ + b^"". 

4. 7^1 ^1 _ mi 71^ x^ 2/1 + m§ ?i? £C^ y^ — 7?^^ ti^ x^ ?/5 + x^ y^ ; 

x^ — x^y^-^xy^-x'^y^ + y^', x-'^-x'^t/-^ + x-^ y-"" 

- x-iij-^ + x-Uj-^ - x-'^y-'' + y-^. 



ELEMENTS OF ALGEBRA. 21 

5. af X* — a» 6A xi y» + a^ b^i x^ y^ — a^ b^ xl if + a? 6i? x y> 

6. a'^(iH'-\-b^; a;*-a:Hl; a:«-a;* + l; ai8-a«6»4-<^". 

7. a;»-x«/ + «*y*-^'/ + /; a;''-xy + y°; 16-4x^0;*. 

8. 16a;*-36a;«/ + 81y^ a;*-^ «« + tV ^'- b^ a;'+ ^i^; 

9. ,,12 _ ,,« // _|_ /,ii 5 «i« _ ai4 ^2 4_ ai-2 ^4_ ,jio ^e _|. ^8 6« - a« ^»^« 

4- '*' b'^ - a^ b'* -h b'' ; ,V a^' - 3^ ^' f + iV y'- 

10. >(•'* — a» ' ^^•- + b''* ; a«-^ — rt^s ^,4 ^ a'" ^»8 — a*' ^" + a^« ^^^ 

- a" &» + «« 6'^* - «* />-^8 + 6«-^ ; 81 x* - 9 a!^ + 1. 

11. .r"' - ./•'» / + x""* i/'-' - x" i/^ + ^12 2/24 _ a.6 2^80 ^ yj6 . ^m 

^12 _ an» + «• ^' - <^' b^ + ^''. 

12. .r3« - x" //" + y»« ; x« - x^^/ + x«« //^^ _ ^so ^is ^ ^24 ^84 

- x" 1/^ + x^- //« - x« //« 4- y/*8 ; yes ; a« _ «« ^4 _^ ^,8 . 

a4 ^« _ ,^2 /,8 ^9 ,^12 ^ ,^^18 ^^24 . ^12^^20 _ ^9 ^16 ^,^6 ^2 

+ a« 6»« m" /i^ - «» 6« wi" 7i« + m" n*. 

13. 2 + ./, 4-2a + a2; a» + i', </«-^»«; 2 - x, 4 + 2x+x'^; 

x«+9, x«-9; a*^ — 6*, a^« + ^/6^;<+ ^^8; 9««^-4^»^ 
9«6_4^4. a2_|_25, aa-25; a»-J», ««+a«i« + i«. 

14. x*-y», x^+x^V+^V+aJ^iy^+.v"; ^w+x, 7«*-7//«x 

+ tTi'^x^—mx*-\-x*', x« + .y«, x^ — i/; x* + 1, x« — 1 : 
a-« + ^-*, a-« — &-•; a*x*' + b*y*', a»x»' — (^« ?/''; 
a; + 2,x*-2x« + 4x«-8x+16; 4a« + 9, 4a''-*9. 

15. 2a — 6, 16a* + 8rtV; + 4an«4- 2«V>8 4-i*; 9a< — 4/>, 

9a* 4- 4ft ; 1 - y, 1 + y + y* + f-\- y* + ?/ 4- ?/; 
oa; + 10, a«x«-10ax4. 100; a^x'^-l, a^x'' 4 1; 
a + 7WX. rf^ — r/'/nx 4- a^ m* x!^ -- am*x^ + m*x*j 
X y — 9 ^, X y + 9 a. 



22 ANSWERS TO THE 

16. 2 a^ _- 3 h% 16 a' + 24 a^b^ + 36 aH^ + 54 aH^ + 81 b^""; 

+ b^""; a^x^" + b^i/^"\ a^x^"" — a^ b'^x^^'y^''' + b^y^""-, 
cxP + b y% c^ x^P — c^ b x^^ y"" + c^ b^ x^^ y^ » — c^ b^ x^^y^ " 
+ etc. * 

17. a;-2" + 2^-2«, x-'*" — a;-2"2/-2"+2/~^"; 2a;y+9, ^x^y"^ 

— 36 a; 2/ + 81 ; a^ f» — b^ /", a^ 2/«^ + ^;« 2/«« ; c' x'' 
+ ^>* ^Z"", c* x^^ — b^y^""', XT — 36, a;-^ + 36 ; a^" — ^2«^ 

18. 2 a;8 + 3 3/'^ 64 a;i« - 96 cc^^ ^/^ + 144 cc^^^/^ - 216 x^ y^ 

+ 324 x^y"" - 486 a^^ y^"" + 729 ?/" ; 16 a;« + 9 2/^, 16 ic« 

— 92/^; a'"6" + ic^/, a^"'¥'' — d^'''h^''x''y'-^a^'^b'^''x^'y'^' 

— or b" x^'' y^' 4- x^^y*' ; l + 2x\\—2 x^ + 4 ic^ — 8 ic« 
+ 16 a;« — 32 a;i<^ + 64 x^^; a"*" — 6«", a"*" + ^«»j 
x-^y-^ + Ij x-^y-i — 1. 

19. ic^r. ^1 _ ^^1, y-j^ ^2„ ^-1 _l_ ^^^-i. a-^jc-i+ 1, a-2^-2 

— a-^ic-i + a-^x-^ — a-2ic-4 + l; fa^ici^H- |^'~^2/"^ 

— i 2/-^ tV ^-^" + ?^T ^-"2/-"' + T^o x~l^y-^^ 

20. ^'s ci« + .3 £ci ?r ^ ^^ ^i'' — 3 ^> cf « ic^ ?/- ^ + .09 b^ c?« a;? 2/~^ 

-.027 ^>i d«a;? ?r ^+. 0081 a^t 2^1; ^16a^ic-f " + .090!-^"', 
16a5a;-l«-.09a;-J'"; 2-t«aTV-3^ i-^,2-§"«V<. + 3^^»-i 

Exercise 30. 

1. 3 a^h-^ — ^^- a + ^V" ^; ^~^ — ^^y~^ + .V"^- 

3. l — 2a — 2aM',xl-\-xh y\ J^ x^Tjh ■\- ?/l. 

4. {a — b — c)"^ — (a — b — c)-2'« — (a — b — c)"". 
'5. 0* + y + ,*; ; a?^ _ 2 a' ?/ + t/^. 



ELEMENTS OF ALGEBRA. 28 

6. x* — 2x^f/z-{- 4y-^2; ar^ -f 3xy + '^^xz + 3/ + z\ 

7. a;-2y + a;ijri+y'. 

8. 2 X*" — 4 x"y" + 2 y"^». 9. a;" y — af " * i/^\ 

10. a'^- - 2 a^b" + 6=^"; a^' — 1 ~ a--^'. 

11. 3a*'' + ^ — 4a«'' + 2a=^«-^ — a«-2. 

12. 2al — 3a->*« — a-i\ 

13. Sx^ -42f-^ + ox'-'^ — af-\ 

14. 2 m'-^ + 3 m'-^ — 4 ?/t'-». 

15. Saf — 4.af-^ -\- Saf-^ — af-^ 16. a"'"— ' — ^("-^>"'. 

17. 2a; + 1, 4a;''^ — 2a;+ 1; 4 + 9 a^ 4 — 9 w^; 4 ^ - 26, 

16a''« + 8a6 + 4 6'^; a + 10, «2-l()« + 100; a^ — S, 
a* -\- S] m — n, m* -\- m^ n + w,^ n^ -\- m n^ + n^ ; 
l--2y, 1 4.2y + 4/; aJ-1, a«6«+ a«6^ + «*Z»^ 
+ ««6»+ aH^H- a^> + 1. 

18. a;=^+a:-^a;2— a:-2; WaX^'x^^— ^^^ul'^h^x^^ -\- ^^aV'h^'^x^^ 

- iT.?«^''*"a:'''+2^?Z>^'; a:^" + y-*"', a:^" — a;*«y-^"' 
+ y-«-; 05' - y^ x^« + a;^^ 5^ + a:^* y" + a:'y" + y«> ; 

3.6m _ 2.8m y8H_,.y«« 

19. 2a« — 3y-», 4 .r< -f 6 a^y* + 9y-«; 4 a* — 3 7i-», 16a» 

+ 12 a* n-« + 9 w-«; ti «*" ar^ + I ^*, 3 aA" + 2, 
81o!- — 54^A'' + SGrtJ" — 24aA" + 16; cx« — ay"', 
c^a;*" 4- ac^x^^y"' + a^c'^x^y^''' + ««^«"y»"' + a*y"; 
J a^- + .04 6J% I a^"- .04 ^J"; 8" a*" + 9", (64)" a^" 

- (72)- a*" + (27)*. 

Exercise 31. 

1. ± 5a;y^; ~2a26a;«; -5«i''; ±3a*i^ 

2. _7aW>-«; ±|ar^y*«*; -a^V; i-'^^ST'- 

3. a:*; ± 11 a;«.y; ±5aft; ±2^-26^ 

4. — 3 a" ^.-*; - 4 7/1 n^ a-«: 7»^ w«. 



24 ANSWERS TO THE 

5. -la'-y-^', 2a3x-2; ±^0"})^. 

6. la^hc^'d-^', ±^0,-^1)"', 2; - 2 a». 

8. £c'"^; 2a2 6*a;«; iOir^^'^+^j _2cc"-2/+s^ 

10. ± 2x''y«"^'^; — fm-^Ti-i; a6^c-\ 

11. \\ a bi c- i, or — \% a b^ c-i. 

^ i_ 13 m — l 

12. V84; 5''xf; 3»aH"; a-i; xy, x » /; icy-^; a^b^^c^\ 

{x + y) (x - yy. 

13. (a h^ c")"; «'» {x — y") ; a'^ x^p ; (a? + ^)2«. 

Exercise 32. 

1. y — 1; 3 a^ _ 2 a — 1. 10. 1 — a -^ a'^ — a^ -\- a\ 

2. 2a3-3a26— 5a62_^76^ '11. Sm-zz + '^^^r + y. 

3. x^-6x^ + 12x—S. 12. x^-3x^y + 3xy^-y^ 

4. a2+2a + 2. 13. 5a;2_ 3 ^^ _^ 4^2 

5. 3 + 5 X — 2 x2 + x8. 14. x^-3x^ + 4:X — 5. 

6. ab — 2ac + 3bG. 15. 2 — 4 ai + 3 M. 

7. 7 a^ _ 2 a - |. 16. ^x^—xy-\-^y^; x^-3x^-2. 
3. 2x + 3y — r)a. 17. 5 a:? — 3 £c* + 4. 

9. m^~3am'' + 3a''m — a^ 18. ip^ _ 2^2 _ 3^-1^ 

Exercise 33. 

1. 182; 6.42; H; ^^; .315; 1.082. 

2. .5555; 75416; 30709. 

3. .2846; .9486; .0316; .3794; .5000; .0169; 1.8034; 

4.5728. 



ELEMENTS OF ALGEBRA. 25 

Exercise 34. 

1. a^^x—1; x^—ax—a^. 7. a — b — 2c. 

2. 2x' + Aax-3a\ 8. 1 — x -\- x^ — x\ 

3. x^-2x-{-l. 9. 2x-^-3a;y + 5y^. 

4. 3a'»-2ci6~^. 10. a + 26-c. 

5. l'x^-l-x-3. 11. x"^ + a;y-2/. 

6. .3x^ — xi-6. 12. 2y- — 3x2/ + 4ar2. 

Exercise 35. 

1. 42; 32.4; .625. 2. ^^, or .0425; .0534. 

3. .861; .430; 2.017; .669; .200; .873; i ^i-i, or .637. 

Exercise 36. 

1. anbi'^crl; 200 a;» (* — .V") (« + JTY (^ — y + «")*• 

2. — 32a\/2axy. 

3. Sx^-2 -\-x-l 

4. 2xi'' — 4 + 3a!r-i". 

5. 4 a;"' 4- 2 aj*" — x'*". 

6. a;iy-J — 2 4-a;-'y*. 

7. aa:»-2 6a!« + 3c. 

8. i x« - 2 X + i a. 

9. rt" Ta;'"; a4±x*. 

10. 2^^ + 4a;y- 3a;«. 

11. x^"" — 2a?^y" + 4y*". 26. 3 a* — 2 a + 1. 

12. X-* — 2ar> + l. 27. x-^ + af — aj. 

13. 2x--iy«. 28. a- 2. 

14. 3a-J — i + 2a-J«. 29. (a + ft)^"* x + 2 a"r. 

15. 1—3 a. 30. x" + x"-* + x«-*. 

16. X— y. 31. 2 — a»"-^ 



17. 


2a + 1. 


18. 


±12; ±8. 


19. 


a 4- 1. 


20. 


a^-ab-{-b\ 


21. 


517. 


22. 


384. 


23. 


a^ — 3 a + 5. 


24. 


X* — (m -I- w) 


25. 


5 a -2^ + 3 



26 ANSWERS TO THE 

Exercise 37. 

1. 4 c. 18. a + 2b — IS G + 7Sd. 

2. a — b + c. 19. — 6 a. 20. 0. 

3. —Sx^ -Sx. 21. 4 a2 _^ 4 ^2 _^ 4 c2 + 4 d\ 

4. —11 X - 2 y, 22. 0. 

5. 3 a - 8 Z> - 2 c. 23. 12 ^8. 

6. — 35 « + 30 ^> — 30 c. 24. — 3 a 7i 2/. 

7. 4 ^ - • 16 ^ — 2 c. 25. 2 ny'^. 

8. X h 2 y. 26. Saxh — 3m + 6 n. 

9. 3 b. 27. 0. 

10. 2xy-- 'J y — z. 28. 3m'^n^ ■\-2 m^ n^ — n\ 

IX. ^^^a + S. 29. ^jy^ + ^xy. 

12. a -- J-^ b -Jr Y- c- 30- S x"^ — 8 y\ 

13. 3 a 4- 4 :?. 31. a (^ + c) + 5 c. 

14. 7x'-hy. 32. (a + /*) — 9. 

15. 210' Z>- 222 a +84. 33. (a; + ?/) + ;^. 

16. ^- xh. 34. (:*; + ?/) — z. 

^^ ^^-^CL' 35. (a + ^,)2_(a+&) + l. 

Exercise 38. 

1 «4 _ j-^ ^3 _^ 5 ^2 _ 2] ; ^5 _ [6 w2 — 3 m8 - 3]. 

2. 3:r-[22/-5^ + 4?^]; »«/>«- [2 a^ ^^ _^ a ^»« - Z.^]. 

3. A:X-\-3ax^—\Qx^ + Bcy~y'\\ x^ — y^—[z^—ab — 3acK 

(3 or - 2 y) + (- {4 7Z - 5 ^}) ; (a^ b^ - 2 a^ b') 
^(^_{ab^-b'}). (4:x + 3ax^) -(6x^-{y-5cy}y, 

5. - [3ay - 2 ab-] - [5 b x - 4. b z^ - \2 c d - 3], 
-[3a2/-2f^^'-4Z*,t] - [5^>ic + 2^cZ + 3]. 



ELEMENTS OF ALGEBRA. 27 

6. - [- a + 26] - [rf - c«] - [1 - ;.] _ [aj + 2y] 

— [/i — 2 //t] — I4:abc — jj], — [2 6 — a — c z} 

— ld-{-l-z] - [a; + 2y-2 w]- [n + ^abc-pl 

7. - pxy-2x] - [5a:«/-4x^y-^] - [xyz-x'fj, 

— [3 j:^ — 2 u; — 4 x-y-] — [5 ic^'y- + xy 2 — x*y^]. 
3. _[_x* + 4«»]-[3a=^-3a^]-[l-a],-[4a»-3a*-x'^] 

-Pa'^-a+lj; - [2 w-4j»] - [37i + l]-[5x + 6y], 

— [2 m-}- 371 — 4/)] — [5x-f 1 -h6y]. 

9. — [rtc — a/i] — [c'x- ai] — [ax + ayj — [3aic — 3x^2], 

— [« 6*— a ?t — a 6] — [c x-\-a x-\-abc\ — [ay+ 2a6c — Zxyz\. 

10. (2a6-3«y-|-46«)-(56x-[-2c'(i-3]). (a-2^* + cz) 

_|.(2_</_l)-|-(2 7;i — X — 2y)-(7i — [— 4a6c-j9]). 
(2 X — 3 X y 4- 4 X- y^) — (5 x^ y- — [x* y' — xy .^]). 
(x»-|- 3 a* - 4a«) - (3^2 _[«_!]); (4 j9- 2m -3/1) 

— (5x— [— 1— 6y]). {an-\-ab—ac) — {cx-\-ax-\-\ay) 
~{h^y~ [jixyz — 'dab c]). 

11. m'-hem*-^?*-!- 12?^i7i- + 8 7i» — 3m2x — 12mwx-12 7i'*x 

H- 3 m x^ -I- G 71 x^ — x^ 









Exercise 39. 






1. 


9; 1. 




3. 4; 2. 


5. 


1. 


2. 


-A; i- 




4. 21; 25. 
Exercise 40. 


6. 


1. 


1. 


16; 9. 


4. 


¥; 1- 7. 4; ]^. 


10. 


3. 


2. 


12; 5. 


5. 


0. 8. 1%. 


11. 


2;-6§. 


3. 


1; 60. 


6. 


Exercise 41. 


12. 


9 


1. 


-1.7. 


4. 


66§; 20. 6. 2. 


8. 


8; 270. 


2 


3 


5. 


5. 7. 5; 9. 


9. 


-t;i- 


3. 


11. 






10. 


csj. 



28 



ANSWERS TO THE 



2. 

3. 

4. 

5. 

7. 

8. 

9. 
10. 
12. 
13. 
14. 
15. 
17. 
18. 
19. 
20. 
22. 
23. 
41. 
42. 
43. 
44. 
45. 

46. 
47. 
48. 



Exercise 42. 

12, 17. 24. 2 at 65 cts., 22 at 35 cts. 

17, 31. 25. 25 lbs. 

$20, $30, $40, $50, $60. 26. A, 60 ; B, 10. 

Father, 48; son, 12. 27. A, 72; B, 24. 

28. 7 years. 

29. 40 miles. 

30. A, 28 ; B, 14. 

31. 103 gallons. 

32. 3.7. 

33. 20, 21, 22. 

34. 24, 25. 

35. 28, 29. 

36. 100. 

37. 200. 

38. Watch, $117; chain, $68. 

39. Linen, $0.32^; silk, $1.95. 



1. 

A, 25; B, 20. 

A, 65; B, 40. 

A, 25; B, 5. 

8, 9, 10. 

2,5. 



5. 

$50, 

Father, 36 ; son, 12. 
100, 200. 
$23.40. 
162. 

First kind, 21 ; second, 42. 40. 11. 
77 at 13 cts., 11 at 11 cts. 
21 three-cent pieces, 18 five-cent pieces. 
Son, $1.04; father, $1.41. 
76 ten-cent pieces, 19 twenty-five-cent pieces. 
$70, $52. 

25 dollar pieces, 10 twenty-five-cent pieces, 20 ten-cent 
pieces. 

Florins, 53; shillings, 71. 

10 shillings, 20 half-crowns, 5 crowns. 

40 guineas, 52 half-crowns. 



ELEMENTS OF ALGEBRA. 29 

49. 4 children, 20 women, 60 men. 

51. Oats, 20; corn, 40; rye, 120; barley, 480. 

52. $1225 at 7%, $1365 at 8%. 

53. f 180 at 15%, $150 at 8%. 50. 30, 10, 223. 

55. Saltpetre, 6; sulphur, 9; charcoal, 6. 54. $5070. 

56. 51 women, 65 men, 150 children. 57. $3.75. 

58. A, 47f miles ; B, 37| miles. 60. 10^ cents. 

59. $313^ at 15%, $16§ at 8%. 62. 7. 
61. Horse, $375; carriage, $300; harness, $75. 

Exercise 43. 

1. 2x2 X<iX ax (I Xbxbxbxx; 2x^XxXXXxXyXy', 

[\X^XaXbxbxbxcXc\ 2 X 2 X 5 X « X ^ X <; X r Xc; 
5x7xa;XiFXa;X2/XyX«X«X«X2X«X«; 
^XlXaXbxbXxXX', 3x3x2x 2xaXbxbXxXxXx. 

2. ^aHxA^a'b', 3x«y3 X 3xV; "d ab'^x'^ y^ X*d ab'^x'"' ^', 

13a4-6i X 13ai"^>i. 

3. 4aU*; 3alMx; 8ai6ic2; 5a-ift-^x»y». 

4. x\'X^x\x^xix^x^\ m'" wJ^.m?", ?Aii''.mi''?/ii''.ml''; 

a;4 . xi . x\ a;i • jc* . a;i . jc» ; xl xl- xi, jc* • a:* • xJ • xK 

Exercise 44. 

1. n{77i + 1); ab(4:a + ftc + 3); 3a*(a- 4). 

2. xia-b-\- c)', x*y^ (39 y» + 57 x""). 

3. x«(5 a; 4- 7); 12bxy^ {6bx - 7 i^ - S axy). 
^ 2 aaf y z {462 at^-^ — 5S9 s:r-^ + 616ay2). 

5. 4 a 6 (a - 1 5 6* + 5 c + 2 a 6» a;* + 4 y - 9 a« c ari) . 

6. a;iy(2 — a6a;l + cajly«); 6a;J(a; + 2a;i — 3). 

7. iac»(i<r-l + a-n-4a4&-lcl); a"a:"(a*-- a"x'' + x2"). 

8. a''^"c*(ft"<r'"4- a'-ft*" — a^^c"). 



30 ANSWERS TO THE 



Exercise 45. 



1. (x + 8) (x + 11); (x - 3) (a; - 4); (a^ - 8) {a* - 12). 

2. {x + S)(x + 27)', {bc-n)(bc-lS). 

3. (aH^ + 12) (a^ P + 2o) ; (« + 11 ^) (a - 6 i). 

4. (6^^-8)(a^' + 3); (^^ + 4) (t/^HH); (aH5)(a«+12). 

5. (ab-]-2c){ab—5c)', (a'' +10) (a^— 12); (n + .5) (71 + .3). 

6. (^'^+762>)(a2-8^^^); (x+5)(x-U); (x''-\-25a'"){x''-12a'). 

7. (i:c-4)(aj-ll); (m + i)(m + |); (oj + 2)(x- 13). 

8. (a& + 5)(a& + 26); (a-5^>:z;) (a- 15^»ir) ; (^/HSa;^) 

{i/-9x^)', {l + i)x) (l + 7x); (m-7a){7ri-Sa). 

9. (a + 9x1/) (a - 21 xy) ; {x + ij + 4.) (x + y + 1), 

10. (1 -5a&)(l-8a6); (a - ^ + 2) (t^ - ^ - 1) 

11. (x-y+2){x-y-b); {x + 21) {x + 21)', {x'-12) {x^-ll). 

12. [(a + Z^)2 + 1] [(a + ^)2 + 8J ; {x^^'-b){x''--m). 

13. (a + 3 ^>2 c) (a - 13 b^ c) ; (ic« + a) (x« - b). 

14. (ic + 5 2/) (a; - 14 2/); {X + 1) (a; - I); (x^" - 20) 

(cc2«-23). 

15. (aj + 1) (oj - 1) ; (x' + 21 C.2) (^2 _ 22 ^2), 

16. (x 7/ + 11) (xy — 14) ; (a" a;^'" + 11 if) (»" :z;-'" -f- 3 ?/«). 

17. (a; ?/ — 11 a'^ 6'*) (a; y — 17 a''^*") ; (a; - ^) (cc - 1). 

18. (a;2»2/'"+17a'"6'")(£c2«^2«_^3^m^.„-). |-(j^_^^)3m_|_7^4nj 

[(aJ + ?/)'"•- 14 a^"]; {n'+ .11) (^i^-.l). 

19. (o^ + f ) (a^ - I) ; (:r + 2.1) (:r - .1); (a' + 1) (a^ + i). 

21. (2 cc - 2) (2 X - 3) ; (3 a: - 3) (3 a: - 6) ; (2 :z; + 6 a) 

{2x + 2 a). 

22. (Sa + 4.b){3a + 6b); {4.x-2a) (ix- 3 a). 

23. (5 x^"" + J a") (5 x'^'" - i «") ; [6 (a - b)^" + 13 (a — b)'] 

[6{a-b)^" - 11 (a-b)^]. 



ELEMENTS OF ALGEBRA. 31 

Exercise 46. 

1. (4x+l)(x + 3); (2y+l)(2y-3); {3a^-\-x') (ia^-x^. 

2. (3-a:)(H-4-c); (4 a; + 3y) (2a; - 7 y); (3aa:-l) 

(2aa:-f 1). 

3. (?/i'-3)(8w» + 9); (3a- ll)(r>a- 1); (2a + 3Z>) 

(3 a — 6) ; (2 m — n) (m - G n) ; (3 a; + 4) (a: + 1). 

4. (8 + 9a)(3-8a); (a:+ 15) (15a:-l) ; (44-3a;)(l-2a;). 

5. (3x~2y)(2a:-5//); (4x-3y)(2x+5y); (a:-5)(15x-2); 

(12 x - 7) (2 a; + 3) ; (a + 3) (11 a + 1). 

6. (3-5x)(6~x); (3x + y)C2x-3y)', (l + 7x)(5~3x). 

7. (8x + i^) (3x - 4y) ; (2 x2« + 7 y*") (3 x'^" - y^). 

8. (x4-y + 2w + 2/i)(2x + 2y + 7?i + 7i); (x + 4)(2x— 7). 

9. {x + y^3a-3b)(2x + 2y-a-b)', (x+|)(Ya:-l). 

10. [(x-y)»--2x"'yi«][ll(x-y)»''-x-y5"]; (3a+l)(9a-l). 

11. [2 a" + 3 (x — y)'""] [2 a" + 7 (x — y)""*]. 

Exercise 47. 

1. ^m2H-7i2)2; (;,i_|.,i)2. (^ a'' - W b cY', a^tt - 2)^ 

2. (7 w» - 10 7i'^)2; (9 x^ y - 7 a')^ 

3. (7ii» - 7*)^ (1 - T) wi7i)»; x^ (x + 1)'*. 

4. (« + /> + 8)2; (7/1 + 0)^ 

5. x*(2«»-5xy)2; (19 a^»c- 2 rf 77i7z)«; (11 7/i7i2_ IOje?)^ 

6. (15x2~y2)2; {2a"'-b'''y. 

7. n«(7 7w + 3^)^ (:^+ i)^ 
8- (s«*+ i*')'; c(aM-3fc»)«. 
9. {3x-},yy', {m--7i-\-iy. 

10. (a2-a + 3)*; (2x + 2y-f 1)^. 

11. (ai - il)«; (;/ii - 1)^; mn (ml - niy. 

12. (xJ + yi)«; {m ni - a)«; (2 xJ + 3 /t)'. 

13. i(a^br-r,rY- {^,xl--^)\ 



32 ANSWERS TO THE 

Exercise 48. 

1. (l-7x)(l + 7x + ^Qx')', {2x-9f)(4.x''+36xf + Sly'); 

(6 x''-a)(86x' + 6ax + a"). 

2. {xy — ah) {x^y^ ^r cuhx^y"^ ^- a^h^x^y" -\- ahx^if \x''y')', 

\x _ 1) (a;6 + a;5 + cc* + a!^ + a;2 + ^ + 1) ; 
(3 a - Z>)(81 a* + 21 aH + 9 a?h'' + 3 a ^>« + h')-, 
(a b"" — m) {a" b^ + aPm-{- in''). 

3. (6a-7)(36a2 + 42a + 49); 8x{l-3x){l + 8x + 9x''). 

4. (a8 - 4 ^2) (^12 + 4 aH^ _^ 16 a« ^>* + 64 a^^*^ + 256 h^) ; 

(9 x-12y) (81 c«2 + 108 ic^ + 144 ^2) ; (^^-i _ y-i) 
(a;-* + x-^y-^ + ^-^^/-^ + x-^y-"" + y"'). 

5. 5ir2(3a;-4)(9a;2 + 12a; + 16); 

2 a6 (a - 2) (a^ + 2 «» + 4 a2 + 8 a + 16); 
(ar-i — 2/~^) (ic-? + x-^ y-^ + 2/~^). 

6. (ab-xy){a^b^-{-etG.); (4.a^-5b) (16 a* -{-20 aH + 25 P); 

(iC" — 2/"^) (^2« _j_ j;c«ym _j_ y2my 

* 

Exercise 49. 

1. (2 a 4- 1) (32 a^ - 16 a» + 8 a^ - 4 a + 1) ; 

(l + ic)(l-a; + a;2-ccHa;^); (ic'+2/') (^'-ic^2/'+2/'); 
(^' + 2/') («' - :e'z/' + ^^^ 2/' — x^ y^ + y^). 

2. (a 4- 2) («« _ 2 a^ + 4 a^ - 8 «» + 16 a^ - 32 a + 64) ; 

(a;2+ Oy) (cc*— 9x2^^+81 ?/') ; (4.t2+2/2) (16a;^-4a;2^2_j_^^). 

3. (ab -\- x^y^) (a^b' — a^b^xy -{- aH'^x'^y'^ — abx'^y^ -\-x^y^); 

(x^ + 4 y^) {x' - 4 x^y^ + 16 2/4^). 

(10 cc + 11 2/) (100 x^-110xy-{- 121 1/'). 

4. (x^ + y^) (£ci2 - cc^ 2/' + 2/''), ^i' (^' + 2/') (^' - a;' ?/" + ?/') 

(a;i2 _ x^ if + 2/1^); 5 x!" (3 ic + 4) (9 tc^ _ 12 a: + 16) ; 

(a;8 _|_ ^8) (^.^16 _ ^8 ^^^8 _^ ^16^^ 



ELEMENTS OF ALGEBRA. 33 

5. {X-' -f y-') (X-* - x-^y-' + x-2y-2 _ etc.); 

{x^-^y'-'Xx'^^-xh/^-y^) ; (xy-^ab){x*y'-abxY-\-etc.) ; 
(a' -I- n (a" - a' i" + 6»«). 

6. (a" 4- *") (a" - a" 4" + i**) ; (1 + x') (1 - X* 4- «») ; 

(af> + y2«) (x2» - x»y2"» + 2^"*); . 
(ar-» + y-*) (x-i - x-iy-^ + yl). 

7. (a*" + 6»"') (a^" - a*" 6»'» + ^"'*); (2a^»c + 3 a;) 

(16a*b*c*—2-iaH'^c''x + S6a'b''c^x^—54:abcx^-\-Slx*)', 
(4a + i-^) (2o6a* - 64a»*^ + 16aH*-4:aH^ + 6«). 

8. (4x'^4-9a«)(16a;*-36a«a;2 + 81a*); 

a «' + J *') (^ «* - ^'ff «'^' + tV ^*); 
(a« + 6 c) (a* - 4 aHc + 11 ^^^c^). 

Exercise 50. 

1. (ax + by) {ax - 5y); (4x + 3y)(4:X - 3y); 

(5aa; + 7 by^) {5ax-7by^). 

2. (ar^ + 5^)(x+y)(a:-y); («^ + 9y2)(x + 3y)(aj - 3y); 

(x* + y*)(ar^ 4- y^)(x + y)(x- y)-, {x' + y)(a;* + f) 
(x'^y){x^-y). 

3. (aH^ + 9a;V)(«'* + 3a:yi) {aib-Zxyi)-, 

(l-\-10a*b'c){l-10aH^c)', (4^8+ 3 6«) (4 a* -3^"). 

4. (3a'' + 2ar»-)(3a"~2x^-); (^ a + ^ 6) (^ a - ^ 6) ; 

(a;J + y*)(a:i-yi). 

5. (a;-«+y')(a;-'4-y)(x-*-y); (a4-^+c4-<^)(a+^-c-rf); 

(x— y + a) (X — y — a)- 

6. (a4-a:—y)(a—a:+y); {ab-\-xy-{-l)(ab-\-xy—l)', 4ab. 

7. (a 4- Z^ + 2) (a - ^») ; (a 4. 6) (a - * 4- 2) ; 503000. 
a 47a;(aj4-2y); 2805000. 

9. 12 (a; - 1) (2 a: 4- 1) ; 1908 X 1370. 

3 



34 ANSWERS TO THE 

10. (a^« + 1) (a^n + 1) (a« + 1) (a» - 1) ; xy{Sx + y) 

(9^2 — Sxy + f) {Sx — y) {^ x' + 3£cy + t/^) ; 
h {a" + h^) (a + b)(a-b)', {ah + bh) (ah - bh). 

11. 6t (a + 4 a; — 6); (2 aj-^ + 3 2/"^) (2 a;"^ — 3 5/-1) 

(4 a;-- — 6 a;-*j^-i + 9 y-^) (4 x-'^ + 6 a-^y-^ + 9 y-^). 

12. 150000; 2{ah-\-2hx){ah-2hx)\ (5a4"+3i^>'«)(5a5"-3i^>'»); 

(a: + y) (a; - 2/) (x^ — a; y + y^) {x'^ j^xy + y^. 

Exercise 51. 

1. (a -^ b + c) (a — b — c)-^ (a -\- b — y) {a — b — y)-, 

(^a-b + 2) {b-a + 2). 

2. {5x-\-b-{-3c){5x—b — 'dc)\ (a-\-x + y-\-z){a-\-x—y—z). 

3. (2a;-3y+9)(2a;-3y-9); (a; + 3) (a; + 4)(a;2-7a;-12); 

(2 a: - 1) (a: - 1) (2 a;2 + 3 a: - 1). 

4. (4a;2+a;-^)(4a52-a;+i); (3a+:r+42/-l) (3«-a;-42/-l). 

5. (a; + 5/ — m + w) (a; — y — m — 7i) ; (a^ + ^^ + c^ 4- ^^) 

(^2 _^ ^2 _ ^2 _ ^2^ . 4 ^2« (^n ^ J«>) (^« _ J«)^ 

6. (;s + 2a!— 32/)(^— 2a; + 3y); (2a + l— 2a;) (2a-l + 2a;). 

7. (x + 3^ — ^) (a; — 2/ + «) (ic + .y + ^) (a; — y — «) ; 

8. (c + (^-3a + 2a;)(2a;-3«— c— c^); {2x—Sy-{-4.z-\-bd) 

{2x-Sy-4.z^5d); {b + c + 2x) (2x-b -c). 

9. (5a«+4a2+a;2-3)(a;2+4a2_5a84-3); (y-^bb + Sbx-^l) 

(3,_5J_3^a;+l) ; a2„^^«.|_2)(a"-2) (a2«_6)(a2' +2). 

10. (3 a 4- * + ^"^ — y"*) (a;" — y*" — 3 a — J). 

11. («» + a^^" + y - 3 ;s) (a^ + xS" + 3 ;s; — 2 ?/2m). 

12. (2aj + 3?/ — 67?. — 4^) (2a; — 3y + 6 71 — 4;?). 

13. (a'' — 6« + c"* + ^^m) (^« _ j« _ c"* — A;2'»). 

14. (2a + 3a; + 4?/ — 8«) (2a + 3a; — 4?/ + 8«). 

15. (a + lb-3c)(a + ^b + 3c); (a^+ a-b"-- 3) {a''--a-b''-\-3). 



ELEMENTS OF ALGEBRA. 36 

Exercise 52. 

1. (3a^ -^ 3 ab + 2b^) (3a^ - Sab + 2b^', (a^ + 3a-{-9) 

(a'-Sa + d); (-ix^ + 2 xy + y^){4x*-2xy + y^). 

2. (x' + xy + y^(x^-xy + y^)(x*-x^y^-\-y*); 

(9 «' + 10 a a: + 4 x^) (9 a^ - 10 a'x^ + 4 x^) ; 
(m* -\- m7i -{■ n^) (m* — mn + n^). 

3. (2 X* 4- 2xy + Sy") (2 x^ ~ 2 xy + Sy^) ; (a* + aH + b') 

(a* - a^6 + 6*); (9 a" ^ 6 a + 4) (9 a« - 6 a + 4). 

4. (5a» + 7aUi + 46»)(5a»-7ai6a4-4^»«); (aj + ^iyi+y) 

(x-iciyi + y) ; («» + a;3 yJ + y^) (a:» - xi yi + y»). 

5. (4a*+4a^^>i + 36»)(4a*-4a'*^>i + 3i»); (3tt^+2a62+76*) 

(3a« - 2a62 + 7^*); (^ + jo* + 1) (/> - p^ + 1) 
(;?«+;? 4- 1) ip" -P + 1) (i?* -i?* + 1). 

6. (7aH4a6 + 96^)(7a'^-4a* + 96*); (3a;H3a;yH5y*) 

(3x^-3xy^ + 5y0. 

7. (m** + m- + 1) (ttj'"* - m" + 1) ; (x^^ + 4x» + 16) 

(x«"-4x"+ 16). 

8. (a 4- a* ** — ft) (a — a^b\ — b)\ 

(a«" + 2 a-6'" - ft^"*) (a*" — 2 a" 6™ — i^'"); 
(5 m* + 2 m » — 4 71^) (5 m'* - 2 m n - 4 7i») . 

Exercise 53. 
1. (a + ft) (a + c) ; (a c + rf) (a c — 2 ft). 
a. (a-.ft)(m-«); (a-ft)(4x-y); a(a + l)(a=' + 1). 

3. (2x~y)(3a-ft); (;, + ?) (r - 3). 

4. (x — y) (a — 2 ft — 4 c). 

5. (a - ft) (5 a 4- 5 ft - 2) ; (2 X + y) (3 X - a). 

6. (2x-l)(x«4-2); (ax - 1) (a»x»-ax -1); (a;-2y) 

{m — n) ; (a 4- x) (4 x — a). 

7. (x + my) (x — 4y) ; (a — x) (4 a — 4 x + 5). 



36 ANSWERS TO THE 

8. (a — c) (S a — b) ; {a -\- b) (ax + b ^ -{- c). 

9. (ox + Sy) (ax — bt/); (m ~ n) (n ~ p). 

10. (m — n)(m + n -p); (2^/ — 3£c) (3?/ + x) (3 i/ - a:). 

11. (c + 7) (3 a - 7 6 - 5) ; (ic - 2 2^) (:r - 3 2/ + 3) ; 

(a, _ 1) (;^2 _^ i^>^ 

Exercise 54. 

1. (a-\-b + cy. 5. (a-b + c- df. 

2. (a-5-c)2. 6. (3 0^2/ -4 ay. 

3. («^ + 6 - c)^ (oj + 2)^; (^ a - 3 6 - t)^ 

4. (a — 3 a:)^ 7. (m — ?i — ^ + a;)^. 

Exercise 55. 

1. 10 ((K" + 1) (af* _ 4); (^2 _^ £c + 1) (a;' - a; + 1); 

12(xy+l)(xy-^). 

2. (a: - .5) (x - .06); (.^ + f) (« + ^); (3 - a:) (2 + x). 

3. 3m27i(m + ^)(^ - m); 2 (2 a - 1) (4.0^ + 2a +1); 

(a^ + 9) (a + 3) (a - 3) ; 6a;«(£c + 6) (a: + 2). 

4. (xy + t'^) (^2/ - t); C^^'^* + <^o) («'^ - f); 

\.(a + by^i]l(a^.by-^l 

5. (a-x)(«'.-a:-4); 2(x'' + x^2){l-x)(2 + xy a(ha'+l). 

6. (a^3+ f ) (a^«-^) ; (a-+o,) (^^-+1) ; (| a' --3 a«) (| a^ m^g a^). 

7. (m-a)(w-7i); (a4.8)(a-l); (a-b)(4.a- U- 2)-, 

(la"" b^^ + a;^) (7 aH^"" + 3yi). 

8. (17 + a) (12 - a) ; (x'- - if) (a;^" - i ). 

9. (tw + w) (m — ?i— ^); «» (a; - 1) (a:^ + 1) ; 

a' (l + b)(l-b)(l-b + b^) (l + b + b^y 

10. (20+a;) (19~a;) ; (a^-l) (8a^-l) (a^"'+ ^3'"+ ^2'"+ a-+l) ; 

l(x — yy^ + 5 ^2-] [(a; - 2/)^'' — 5 Z'^'"]. 

11. (3a;-ll)(2a: + 7); 12(a:+7) (aj + 2); (x + y) (x + 7j-5); 

(f a: — J^??^7^) (fx- ^ ?/). 



FXEMENTS OP ALGEBRA. 37 

12 (H-^3^ar)(l-rVa:); (2ar-2/)(ar + 3.v-2a). 

13. (a" + b'x) (a^ + />";/); (a- - (i) (a; + 2« + i) ; 

(9x-^ + liajy - y-) {'^x' -2xy- y«). 

14. b (a* + ^^) (</« -aH^ 4- 0') ; (a; + y)»; (x^" + aZ» + ac) 

(ar-^" — w ^ + ^ c) ; (ni -{- n) (m* — ?/i w + /i'- + 1). 

15. (rt + ^— c— r/)(a-/> + c— </); (a— y + a; + 2) (a— a + y+2). 

16. {x -^ 0) {x-\- a-i- b)] {x^" + 2 a) {x^" — a — b). 

17. 2 (57/i + ii^ ± 1) (25r?iH 50//in + 25w^ if 5m qp 5w + l); 

(4w + 70(4w-+2m7i+77*2'), 3w(12w^+8mw + 37r). 

18. (^c*af — a*) (/•'•(j^x" — 1); {7 p'' + 13;;^ + 11 q^) 

{Ip'- V6p n + 11 q^) ; 2 m {m'-\- 3 7^, 2 w (3 t/i'H n^). 

19. (m— 7i) (2m— 271 + 1) (27?i— 271— 1); (m + 7i) (2??i + 7i), 

71 (m -f- n). 

20. (a- + a c + /> c) (« — a 6 — « c) ; (8 m* — 4 m ri + 9 n^) 

(8mH4m7i+97i2) ; (r)a;2+3xy+4y^) (5a:2-3a;y+4y2). 

21. x(3a; + 2i/)(2a; + 3y); ^/'(Sar + 4y) (2a; - 3y). 

22. (a:» — 3ic)(a:" — a^» — ac). 

23. {m^^-\n^){m±2n)\ (4m + 37i - 3j9) (3w + 3/> — 2m). 

24. (x»«" - c) (x"" + a + i); ' (X + y + 2)(a: + y - 3); 

(6ar» + 4 a; y 4- 4 y*) (5x^-4 a; y + 4^). 

25. 3xV(3« + 2y) (a; — y); (m - 37?.)(m + 2 ti ± 4). 

26. m\n\{ab—xy--^z)(,iv'b''-^abxy-\-?^xyz-\-?>z^-\-x'%f)\ 

(9 a- -f aJ" b^"^ — 11 i'") (9 a" — «»" ii*" — 11 A"*) ; 
(9 a*» + 3 a-i^™ — 5 ^»*'") (9 a«" — 3 a» 6^*" — 5 i*'"). 

27. 2(3x-2y)(3x-2y±6); 2 (m + 37») (m-27t- 6a); 

a«(a''x^+4n^) («*x*-4a«7i2x-^+16a*7i^x*), a^(ax-}-2n) 
{ax—2n){a^x^—2anx-\-4?i^)(a^x^-\-2ftnx-\-4n''). 

28- (x + 7y)(aj-4y + 4); 2(1 - 3 a - 2^) (// - x). 

29. 7»77 (m + 7i) (m — ti)*; — a (a + m) (a* -\- 2am -\- 2m*). 



38 ANSWERS TO THE 

30. (3x — 4:y)(5x-\-4:y — 5a); (a — b) (a — c) {c ^ b). 

31. (a -h c)(c — a)(cd — l)(c^d^ + cd -{- 1); (mn ± 8) 

(w^Ti^ qp 8mn + 64); 6 w^ (4m + 3 w) (w — 2 w); 
(a a; — 3 Z> y) (a — i/) . 

32. (m - 2 7i) (m - 3 w + 16) j a;^ (3 a; -- 1) (a; - 1). 

33. (m — n) {6 m^ + 5 mn + 7 n^). 

34. 9?«.'(a2+ w^), 9m'(a + m)(w— a); (x+42/)(x — 42/ + 1), 

(x - 4 y) (a: + 4 2/ + 1 ) ; (a: - 2 a: 2/ + 2) (a: - 2 xy - 3) . 
Ten. 

35. (36a:-132/)(18ar + 29i/); m(m+ 1) (m-1) (m^-w^-lO). 

36. vi{m-\-n){m^ ■\-mn-\- n^){m^—mn-\-'nP)) {y-\-\){x—l) 

ix-y^-l); {x-\-2)Hx-2Y. 

37. (?/i + 4)(m4-5)(m — l)(m — 2); (3 — w)(ww — w — 3). 

38. (x» — ^ — a:-")2; {x-'^ -^ y-^) {x-^ — y-\). 

39. 7o2a:(a;-2a)(2a:-a); (x-«+y-«) (a:-* + 2r*) (a^*-2r*). 

40. a:2(12a:8-8a'2/H2l2/); a:i(4arJ±3) (16a:q:12a:i+ 9). 

41. (a: - y) (a; — 2 ?/) {x + ari^/i + 2/)(a: — ariyi + y) ; 

(a;- + 1) (x"* + 2) (a:- -f 4) (oT + 5). 

42. (2a + 3^') (2a-3/>')(ar-2a)(a:2 + 2ax-f 40^^); 

{m + 2n-^p)(m + 2n —p) {p + m — n) {p—Di + 2w) ; 
(a:- + i) (x- - i) (a:^- + ^V) {^"" + tV); («'" + *'") 
(a*" + 5») (a*" - b^) (a*"' - b*'' - 6 a^"* ^»2«), 

43. (x^ + 2/') (a: -- 2 I/) (ar^ + 2a^y 4- 4 y^) {x^ - x^y''^y')\ 

(a;"* + 1) (ar^"* + 4) (a:^"' _ ar« + 1) (a:*"' — 4 ar^*" -f 16). 

44. (^-\-n— p){m-\-p — n){m-\-n-\-p) (m — n — p). 

45. 4 (a:"* + 2) (2/" + 4) (a:- - 2) (i/" - 4). 

46. (a:"* + 1) (a:-" + 2) (x^"* + 4) {x^ - 2) (a;^"' _ a:*" + 1) ; 

(2 a:"* + 3) (4 x""^ + 9) (2 a:"* -3) (a;"»- 1) (a;2'" + a:"» + l). 



ELEMENTS OF ALGEBRA. 39 

Exercise 56. 

4. 6a«*x*; 2axy. 13. x {x - y). 21. 4rz-l. 

5. Za'^^y^ !*• ^-^^y + y''' 22. x + 2. 

6. 6x^2/-^z^ 15- 4(a-^). 23. x - 3. 

7. 2a:iyJ. 16- 4;s(a:-y). 24. m - n. 

8. 6(« + ^)- ^^- 2x-3. 25. or -2. 

9. ^y^^. 18. x^'Sy. 26. x^^-e. 

10. a:*(3x+2). 19. x - y. 27. a:" - 5. 

11. 3a«x-12a». 20. x^ - x. 28. 2a:"-5. 

12. x + l; a:" + 6; x + 3. 

Exercise 57. 

1. ar« _ 3 X 4- 2. 9. ^^ (^ — 3). 

2. a:^ — 2x + l. 10. 9m»(wi-l). 

3. ar» + 2 X + t. 11. a: - y. 

4. (x - 1) (x - 3). 12. mn{x^- 3). 

5. x{x- a). 13. x^ - X — 1. 

6. X — y. 14. 2 X — 5 y. 

7. X* + 2 X + 3. 15. 2 n (m^ + 4 m y + 7 y'^). 

8. 3 X + 2. 16. 2 m" x" (x« - 1). 

Exercise 58. 

1. 2(x + y). 3. X4-2. 5. x" - 2. 7. x-2y 

2. ar» -f X + 1. 4. 2 X + 3. 6. 2 x^ + 5. 



40 ANSWERS TO THE 







Exercise 


59. 


1. 


x'-Vy. 


17. 


x'' - 1. 


2. 


x-y. 


18. 


71 (n + x) (w — x). 


3. 


x-1. 


19. 


x — 2m. 


4. 


x-1. 


20. 


xyi^-yY' 


5. 


x-'d. 


21. 


a-b. 


6. 


'6x'- + 1. 


22. 


a -{■ b + c. 


7. 


x^ + ^y + /. 


23. 


X" + 2, 


8. 


x+l. 


24. 


x-2. 


9. 


x{x-^ b). 


25. 


2{x + y). 


10. 


2x'--^. 


26. 


7^2_^8x.+ l. 


11. 


a" - h\ 


27. 


n + 2. 


12. 


a-h. 


28. 


3m(y'+4f-2y + S). 


13. 


3a:2« + 2m2. 


29. 


x''-Sx-\- 1. 


14. 


(m -n)(x- y). 


30. 


2(:r+l). 


15. 


x^ + A. 


31. 


x--2y\ 


16. 


a" + b"^. 







Exercise 60. 

1. S19axU/z^ 10. (n-xy(n* + a^x'' + x^). 

2. lUm^n^x'z^ 11. x^-6x^ — 19x + S4:. 

3. aea^^-^cl 12. 105xy^x^-y^). 

4. 72 m^n^y^. 13. :r^ — 1. 

5. 12aic3y4(a?2-3/2)2. 14. (3a:+ 2)(a^ + 2) (;r+ 3). 

6. m'^n^ (x^ — y^). 15. {a ^ x){b -\- x) {c + x). 

7. 12 a:iy2 (a,2 _ y2>) j^g^ 3mn {x - yf {m - n). 

8. (x2-16)(:z;2_25)(^_6), 17. („4.^)2(^2_^2). 

9. x(x + 2y{x+l){x-\-'S). 18. ^i^-l. 
19. (a:^-/)(x2_^2). 



ELEMENTS OF ALGEBRA. 41 

20. 3a«x(3a:-a) (2a: + 3a) (a: + 5a). 

21. (x + 4) (X + 3) (X + 1) (x - 2). 

22. (x-y)(3x-2y)(4x-6y). 

23. a* - 1() 6*. 

24. {a -I- ^) (w + w) (x + y). 

25. (4 a; - 5y) (2 JB - 7 y) (x + y). 

26. (x^ + y^) (x* - x^y*^ + /). 

27. 20x2y(3x+l)(5x + l)(4x-l). 

28. (a-f ^ + c + d)(a + ^' — c — c?)(a4-c — i— e/)(a + rf— Z» — c). 

29. x* + xV + y*- 

30. 6x2(x + 7)(3x-f 5)(3x-2). 

31. 12x-(x" + 2)(2x« + l)(4x"-7). 

32. abc(m — n). 

33. (a — b){b — c). 34. ale {a — x) (^» — x) (c — x). 

Exercise 61. 

1. 2x*4-x«-17x2-4x-f 6. 

2 and 3. x* + 5 x* + 5 x'^ — 5 x — 6. 

4. (x-2m)(x + m)(x* + m*)(3x2-mx + m«). 

5. 2xy*(30x» + 95 x*y + 68x»y» + 32x«y« + 24xy* - 15y»). 

6. X* - 14 x« + 71 x** - 154 X + 120. 

7. (x2-3x + 2)(x* + 3x*-8x« + 40x-96). 

8. ar* + 2x»-9x^-2x + 8. 

9. 3(6x* + ar*~33x» + 43x«-29x+ 12). 

10. 6 X (x - 1)* (.r + 1)«. 

11. x*4-5x»4-5x* — 5x — 6. 

12. 2x>-2x'-3x« + 3x*-2x«-3x'' + 2x + 3. 



42 ANSWERS TO THE 

Exercise 62. 

1. (a - ft) (a 4- by (a^ - 4:b^) (a^ - ab + b^). 

2. x^" + 7 x^« — 10 x^'' — 70 a;2" + 9 a" + 63. 

3. (a;" 4- 4 2/"') (x" — 2y^) (x^" — 2a;"2/;« + S?/^'"). 

4. 2(x + 3) (2 X + 3) (x^ - 1) (x^ ^x^ + l)(x + 2). 

5. xy{a^x)(b-y){2b-y)(2b-'-xi/). 

6. a*"* — b^"". 7. x*" — 16 a**". 

8. (ic" + c) (2 x" — 3 ^») (a;2» -j- a a;" — ^»2). 

9. (ic + 2y (x' + 4) (X - 2) (x ~ 3) (x^ - 16). 

10. (X2« ~ a^) (aj2« _ ^2) (^2« _ ^2^ (^6« _ ^6-^^ 

11. Sx — y, (3x — y) {x + yf {x — yy. 

12. 3^2 - 2 < 3 ic^ y (3a;2 - 2 a^) (a; - 9 a) (2 x + 5 y). 

13. 2 a;" + 1, (2 x« + 1) (x^" - 1) (9 x""^ - 4). 

14. The expressions are prime to each other, (a* -\- a^b^ -f- b*) 

(a + by (a - by. 

15. a; - 5 ^>, 6 (a; - 5 ft) (x^ - 9 ««). 

16. x^'* — 7 a;" + 12, (a;2« - 7 x" + 10) (a;^" — 7 a;" + 12). 

17. c a; + *^ (c a; + ^^) (a^^ - c^)- 

18. m^ + a; 1/, (w'^ -\- x y) (4: x^ — 9 y^). 

19. a^m ^ ^2n^ (^2m _|. ^2 «^) (^2« _ 4 ^2m)^ 

20. a;^ + X 1/ + y^, (x* -{- x^ y^ + y*) (x^ — 4 y^). 

21. 5a;2 - 1, (5a:2 - l)2(4a;2 + 1) (5x' + x + 1), 

22. a;" — y"*, (a;« — .?/"•) (a^^" + x^^i/^'" + y*-^), 

23. a;'' - 8 a;* + 50 a;2 - a; - 42. 

24. (x'' + 7x-^ 12) (a;2 + 0! + 3) (a; - 2)^. 

25. (2 a;* + 5 ^2 ^ 3) (4 x* - 49). 

26. x(6x^- 31 x* — 4x^ + Ux^ + 7x- 10). 

27. a^s + 3 a;^ - 23 .t^^ - 27 x^ + 166 a; - 120. 

28. 3 a;^ 4- 2 a;^ - 3 a! - 2. 



ELEMENTS OF ALGEBRA. 43 

Exercise 63. 
2 2x — 3y 



2bxy' 37/i«aj«/' 2x ' 

« . n(x* — y^) 2 X + 1 

m X — J 

^ 3m + 2 .. ^ y"-» „ * + a? 3 + a 

^- 3^;^^ ^(^-y>^ ^- °- r-f^' -2- 

3m — 4 3(w + w) a + i + c a — ft — c + x 

4m — 3' m — n ' a—b-\-c^ x — a-{-b — c 

3 x — b m^ + n^ x"*"^ 



4m*(l— x)' x+c" * m * b(a + b) 

7n^n-\-x X — y x — 2 5x^4-1 

X ' x + y ' a + 4' 9 x» — 4 X ' 

x—y—m c — d a — 2 b 2x^^ — 1 



Exercise 64. 
^- ^ + ^^ + nr3^' ^^^- 2x»-x-3 ' 

3. x« + 3ax + 3a''+-^^; a; + 1 ^ "^ ^ 



X — 2 a' ' x*4-ar-12 



4. (x-y)«; x + m + 14-^^— -^' 

X -|- n 

X — 10 

5. 3x«-4x + 5 + ,f— i^; x'^ + y*-. 



44 ANSWERS TO THE 



Exercise 65. 



2m IIP' ^ 2(^H1). 2m« w^ 

' ic + 1 ' m^ + ^^' 2^ — 2/ 



2. . ; — ii— : : ;;• 5. — -\ 



6. 



a + ic' 


m — n^ a- 


-d 


m« 


2y« 




7/1+^ = 


' a:Ha;y + / 




a 


2 




mn^ 


m — 1 




3.2. y2„ 





7. ^i 7- • 8. 



(m + w)2' (m - /i)^ 
2/' (3 a;^ - + //") 



^,rn ^ ^,«y. ^ 2/'^» "• ic'" + 2/" 



Exercise 66. 

n — m n — m —(n—m) n — m h — a 

' h — a a — h ^ — Q) — ay —(b — a)^ x — m—n 

b — a — (b — a) b — a 

, +-7 -x; etc. 



2. 



m+ii—x^ —{x — m—nY —(x—m—n) 
m — a m -\- a — x n — a — b 



n — b^ m — b -\- y '' m — b -\- a 

m — x m — X (m — a)(m — b) 

n —y ' (jn — y) (n — z)"* (m — c)(n — x) (m — y) 

2a: — y — 3 a — c + 3 

4. ^ ■ 



5. 



{a + b')(a — m) (b — 2 x)"* (a — c) (m — n) (x — y) 

ab(m — x) 
mnxy (a — b) (a — c) (b — c) 

—^y{^ — y) 

abc{a — b){a — c) (m — n) {x — y) {y — ;*;) 



ELEMENTS OF ALGEBRA. 45 

Exercise 67. 

amny bm*y bmnx ab^y ^ c 2 b 5 a 
bmnybmny^bmny bmny abc^ abc* abc 

cm-\-cn am— an bn . Sa^bm 3a^m Sd'b — Sabn 

abc * abc ^ abc 3abm Sabm Sabm 
bmn 
Sabm 

10 m n-\-20n^ 10 m^ — 15 mn 15 m — 3n 
30 m 7* ' 30 mn ' 30 mn ' 

a ;* -4 (x-^l)(x^-^) (g - 1) (a*- 4) 

♦• (x2-l)(x^-4)' (x^-l)(x^-^)' (a;«-l)(x^-4)' 

(m — 7iY (m + 2n) {m + n) m' (q -- 6) (a ^-f ^^ 

2 (g + ^) (<^' + ^') 4 (g* - 6^ 
a* -6* * a* -6* * 

120 m -f 30 10 n — 5 9m-6 

*• 16 (w - 2) ' 15 (m - 2) ' 15 (m - 2) ' 

7 2a«y m« - n« 

2 a; (x — y) (wt + n) ' 2 a; (x — y) (m + ») * 

m (m' — m X -f x^) n a (m + a;) 

m« + x« ' m« + x« ' m» + x« * 

• a^ + xV + y*" a;* + x»y« + y*' a:* + x«y^ + y** 

10 ^ (^ •- y)* 5(^' + y') 5y(x^- K.xy4-y') «' - y* 

• 5(x»-2^* 5(x»-y«)' 5(x»-y«) ' 5 (x» - y») ' 

X" -i- //" x^yCx'^-y*") {x^^ -t- y«")« 



46 ANSWERS TO THE 

a{b — a + x) (x — a — b) 

(a -\- b + x) (a — b -\- x) (b — a -\- x) (x — a — b)' 

b (a + X — b) (x — a — b) 
(a + b + x)(a — b + x) (b — a -\- x) (x — a — b)^ 

x (a + X — b) (x — a — b) 
(a -\- b -\- x) (a — b -\- x) {b — a + x) {x — a — b)' 

— a —b c 



13. 



(a -c)(b--c)' (a - c) (b-c)' (a- c) (b - c) 






x—1 6—2x 

15. 



(a, _ 1) (a; _ 2) (a; - 3) ' (x - 1) (a; - 2) (a: - 3) ' 

9-3a: 4 a; -12 

(a._l)(a;_2)(a;-3)' (a; - 1) (a; - 2) (a; ~ 3) * 

mx — am x^ — nx 

{a — x){m — X) (n — x) (« — x) (m — x) (n — x) 
ax — am 



17. 



(a — x) (m ~ x) (n — x) 

a;2-2a;-3 - (2 -f a;) 



(1-a;) (2-a;) (3-a;) (5-a:) ' (1-x) {2-x) (S-x) (5-x) 

-(a; + 3) (a; -3)3 (x' - 4) (a; - 3)^ 

^^ (x^ _ 4) (x^ _ 9) (a; - 3) ' (a;2 - 4) {x^ - 9) (a; - 3) ' 

•- (■T^-16)(a; + 3) 2 (a; - 2 ) (x^ - 9) 

(a:2-4)(a;«-9>(a;-3)' (a;« - 4) >« - 9) (a; - 3) ' 

jB»"»4.3af" a;*"* — 1 a;^"* — 1 

^^' (a:*'«-l)(a;«'«+3)' (a;*'»-l)(a;^'" + 3) ' (x*'"-l)(a;'^'"+3) * 



ELEMENTS OF ALGEBRA. 47 

Exercise 68. 

6a»-16a-15 . IS b^ c +1S b c^-\- 9 a^c-\- 9 ac^-SaH+ Sab* 
^' 36 a ' 72 abc 

12 g» 4- 28 a;' -27 . x* + y* Aa + b 

^' Sx* ' xV ' 3b ' 

cp-\-bm — an Sam ■\- 2han -\- Ibbn 
abc 12 an 

9 a + a' + 12 . 6 (n — m) ^^ a — b 

4. n > rrz ' 6. 



7. 



3an triTi n 

a«-3a6c + 6» + c* . a^' + a'c'-^*c« 
a 6 c a^h^ c^ 

2a^ 2w^ 

13. 



8. 


4 n — w — 


^=' + «^ 


mnx 


9. 


31 47 


19 


16x ' 42m 


30 y 


n 


a«6 + ft«c- 


• ac* 



14. 



11 



ar* — y^ ' a;* — m*a5 

m + 4 -2 

m — 4 ' 4 ?«,* — m 

m4- w, 2ax 
abc '^"' m — n 8a;' — a* 

1 . Sx m — 7 

' ar»- 9a; 4- 20' ^^^^' 4m(m«-3m + 2)* 

4a — 66 5 

"• 3(m«-n«)' (a;-2)«(aj + 3)* 

16 0- ^^^ 

' {m -{- n + x) {m -\- n -- x) (m — n — x)' 

17 ^^y* . 1 -^ 2m^ 
•'^- a;*-/' x»-y*' ^'* m« + n»* 

g'+ 7 2 m* — 2ma;« 

**• ar« + 6x + 8* ^- {m^^x^Y ' 



48 ANSWERS TO THE 



21. 


a; 




27. 


^' + ^' ; 2y. 


l-a;« 


22. 


96 x^ 




28. 


1 


(3 + 2 x) (3 - 2 x)^ 


X-1 


23. 


0. 






3^2 _ ^2 _ ^2 _ ^2 




1 




29. 


(^ - a) (X - b) {X - 0) 


24. 
25. 


X + y 
-1. 




30. 


c — a — b 
(a-c)(b-c)' 


26. 


1 




31. 


2x» . Sa + x 


2x + l 


x^ — 4 ' a + X 


32. 


1 


0; 


2x2- 
64 


9a; + 44 33^ ^^ 


(rn^x){x^2)' 


+ x« 






Exercise 69. 


1. 


J3 ^ 3.m + « 




8. 


m {m — n). 


' 10' 2/'" + "' 


2. 


8 4cie* 




9. 




3. 


1 . 3a^ 


-1 

-2 


10. 


x^ a^ y"- b"" 
a^"^ x" V 2/'' 


2 - X - ic^ ' X - 


4. 


aj+l . ^ 




11. 


(a _ c)2 - 6^ 


x + 5' rr^ — mn 


4-71^ 


a6c 


5. 






12. 


1 


6. 


m 




13. 


a;«-l 


m — n 


a:«+l 


7. 


m^ — n^ . X + 




14. 


^2n _ y2m 


2 (a;2» + 2/''-) 



ELEMENTS OF ALGEBRA. 49 



Exercise 70. 

acmx 3 4 xy . xy 

3(a-^b)\ a*-^2x^y-\- 2 xy' + y' 

b{a-{-b)' x*^3z»y-^4x^y^-Sxy^-{-^' 

x + y . 1 ^ b — X 



m*— 2m4-4 sb*— y" * a — x 

2 a; — 1 g + 6^ — c 

**2x — 3* ' a-\- c — b' 

"• »* - n*x^'+x* ' a^ + a-' + l 
12. 2a;V-4a;V + 2xV. 



Exercise 71. 
a^-b^ 10. 1. 13. 1. 



a — 12 >! «rfi+i 



(J a: - y)« ^ , a "* « 

3. i ^lJ- . 4. X + 6. 

x-y 1 1 

5 (a + ft) (g - 2 y) X x* 

• (a -I- 1) (X + y) ' „ J 

2a6 b a 

7. 1. 8. 1. 9. 1. 1^- ^'-2+ -,. 

^ m5 77tt-r,r-^; 8^^-«- 17 ^^ . 18. lOa. 

h^ J8 6-y" ■*-^- aP^c* 



50 ANSWERS TO THE 

Exercise 72. 

03 + 6, a ■}- ni ^ an -{- cnx ^ mnot^y — 3ic^ 
ic — 6 ' h — m^ cm + cnx^ m^ n^ •\- 2 in n^ x 

h ^ m -\- n ^ 7ri^ — h n ^ 2 m 
a ' m — n a 7i ■\- b in m — n 

x^ + 1 x'^-U 1 

3. —^ ; -t; T^ . 5. 



2x ' x^-lU - 2j(rn-n-p)' 2x^-1 

.. . r (a+x)(a '-x') x'+l . -, x'-x+l 

1 + a;2 a;2 _ 3 ^ ^ 1 4 



7. 



8. 



14-a: ' ic^_4£c + 1' 3(1 4- a;) 

* + 2/ . am-{- adn _ £C — 2/ 

y ' bni + cn-\-bdn x + y 



m -\- n wn 

9. a + a;; 1. 10. -2- 11. • 

{m — n)^ m -\- n 

Exercise 73. 

^' ~8F' 16x''y^'' x'2/ ' 256x12* 

160^"^^. (a; + yY . rn'ix-ijY 
^' Slm^nf^' {x-yY' n'ix-^-yy' 

{x^ - i/y . (a - bY . a''(a''-15a^+75a-125) 
*• (m + nY ' (2c^ + 3 6)2' ic^^ 

(«_5)2 8 a;'^ - 36 a;^ y" + 54 g;B y^» - 275/»" . /^y'+" 

5. . . ,..; Tin » I ; 






_ r ai m^ x^ 1 ' 



ELEMENTS OF ALGEBRA. 61 







Exercise 


74. 






x^ Sm*n* 


6ac» 


«* 


m 




1. 


^^*' X ' 


7x» 


' 36*-* 




2. 


a»» ' b^ ' 


4m?i^ 








5aH* 




3. 


L*a*a''- a~* . 


aixi 


5. 


x + 1-^; a« 


-hi 


3i/-z' V^' 


4. 


aj + y . / 


.^7 * 


6. 


a ft. 3 


-r*'- 


2V(a;-y)-' ^ 






Exercise 


75. 




1. 


1 . x-S 

1 + x^' x + S' 




5. 


46; g. 




2. 


Ty^ — Axz 




6. 


]§. 8. 0. 


10. -H 


6««-7xy* 






*; 0. 9. 8. 
a 






Sx*y-2» 




7. 


11. -f 


3. 


9 x« - y«« * 




15. 


(a - cy - b\ 


17. 1. 


4. 


aj + 5 




16. 


a:2 4- 3 x + 3 - 




12. 


":+"%.»; 


^:-i 


+ K 







13. m« + — 4- ^ + i; :l _ ::^ + !^^Jf _ '^, 
n^ n"^ v}' y' ny"^ n'y n» 



52 ANSWERS TO THE 



18. 



19. 



\n inl \n inf \ a/ \ a^J \ic** y^l 

(a^ ax x'^\ 
l?~b^'^ yV '■ 

5^(y'" + l)(y" + l)(2/" + l)(y"-l),orx*»(2/^"+^) 



20. ic. ^ "~ y ''®- ^- 

2^ 28 (a: + 4) ^^' ^2/ * 3^ ^ 

36. ^-±^ . 41. 3. 

^c + ac -*- a6' 42. 1. 

2b (by — ax) 



22. 



23. 



(«,_,)(5_c) 
24.-^. 

25. 1. 28. 0. 30. 1. 

26. 2. 29. a;. 33. 3. 

27. 0; (a«-68)2. 34 2. 

a:(a^ + l) . 1. 1 45. '^ 

^^' x' + ^x + r ' («^ - ^0 (^ - ^) 

32. '^; ««^'^c^ 46. 4- 



o#. 


ax (S ax — 5 by) 


40. 


16 aH^ 


43. 


2{a + b + c)^ + a'' + b'' + c'', 


44. 


1 


c (a — c) (b — c) 


^1; 


2 a 



ELEMKNTS OF ALGEBRA. 58 

Exercise 76. 

1. 10; f. 4. 6; -|. 6. 2. 8. 4. 

2. 20; 5. 5. i. 7. 2; 6. 9. -2; 14. 

3. 2. 

Exercise 77. 

1. 1; 6. 5. 8; 0. a 4. 11. 3. 

2. 1.3; -2. 6. -4. 9. 4. 12. -.04; i. 

3. 0; f 7. 1^. 10. %. 13. li; 3. 

4. 2; 0. 

Exercise 78. 

1. n — ;— ; wv(l-3a). 5. a-b; — — r 

2a 2ah ^ n * 

^* 36 ' a + ft <? 

1 ft I ^ 

3. 4m + 8n; -T- 8. 0, -; ft-1. 

'aft c 

a • w» 

4. j^. 9. i^ + 27^- 

Exercise 79. 
^ c , ,, 13. H« + ft + 3). 

3ft 2a» 

2. — • 3. 4. ^^ 28«+5ft«* 12. 5. 

5 ^(a-6+c). *• ^* 16. —mn + mp-{'np.l^. 0. 



6. 0,-; 17a. 7. 3(n-l). 



11. 2. 



a 
a* 2 oh 



o 



b — a* a + ft 



6. - "^ . 3 rf» _ 2 c« 17. 36. 

21 19. — To— TJ— • 
12 c a 

7. 0. • 18- ^• 



1 n (n^ 4- m') 



54 ANSWERS TO THE 

Exercise 80. 
2. 31. 3. 84. 4. 36. 6. 1^^ days. 

abc 



ab -{- be + ac 



days. 8. 10 days. 



2abc . 2abc . ^ 2abc 

9- —r-, 7-T-' -A., —^ 7 days; B, , , , 

ab-\- ac + bc ac + bc — ab "^ ab + bc — ac 

2abc 

days; C, -^j—, r- davs. 

-^ ' ^ ab -\-ac — bc 

10. 48 minutes. 12. 16 miles. 13. ^ miles. 

b-{-G 

15. 150 miles. 14. 742^ miles. 

16. 56 hours; 84 and 70 miles, respectively. 

acn . . acm .. _, abn 

17. - — ; hours: A, - — ; miles; B, miles. 

bn + cm bn-{-cm bn-\-cm 



18. First kind, — ^^ ; second, — ^^ • 

m — n m — n 

19. ^^. 21. 69. 24. 160 miles. 



2 m n {2 m -\- n) 
4: m^ ■\- 4: m n — n^ 



20. -, — 2—-^, =^-^ days. 



22. $1200 in 5 per cents; |2000 in 6 per cents ; sum, $3200. 

$100 ^•m . ^100 b(n-m) . ^ 

23. ^ in a%; — ^ ^ in c%; 

am -\- en — cm am -{■ en — cm 

$100 b n 
sum, 



am, •\- en — cm. 



ab -1 , a(3w2_^3w + l) 
25. miles an hour. 26. — ^^ ^!^ — — - . 



b — ac (ti + 1) 



ELEMENTS OF ALGEBRA. 55 

Exercise 81. 



(a: = 24, I « = H 

i3, = 12. "• 13,= 7. 



3. }^ = 2J a. J^ = 24. 12. j^ = l^ 

Exercise 82. 

*• ■jy = 3. "• (y = 12. " U = 3. 

_ (« = 3, (a; = 38J, ja; = l, 

'■ ly = 2. 1y = -21J. • (y = 4. 

6? 
SI 

1. 



'•iy = 60. *• iy = -6. "•iy = S 

ty = 12. '• |y = 7. ly = - 

,x = 19, 



r X = i», 

1y = -i- 



1. 



Exercise 83. 
x = -t, . (1 = 21, „ f« = 13. 



)y = -i. 1y = i5. ty = ll 

a. J'' = J' c-r^fi 10. 1'' = ^' 
(y = 2. (y = 6. )y = 5. 

fx = -3, <x = 10, 11 j* = 4' 

'•iy=12. '1^ = 11. "•1.'/ = 6. 

4 (* = *• 8 i*= ^' 12 i' = 216, 

'••ly = -2. "•ly = 12. ''•ty = 144. 



56 ANSWERS TO THE 
Exercise 84. 

^•iy = .02. "• 1^ = 12. '^- 1^ = 12. 

3. .P = -2' 15. r = 12, 27. j^ = 8, 
b = -3. |y= 6. (^ = 7. 

fa,= 10, ra; = 7, p = 4^, 

ly= 8. U = 5. ^"- t2/ = 5i. 

5. 5=^ = 25' 17. 1^ = 10' 29. r = f 

«-L=4: ^°-{,=i5: ='='-i,=io. 

^•{, = 12. ^^•{, = 1. ^^■], = .08. 

f* = ^' 22 jx = 20, (a; = i, 

iy = 2. ""• 1^ = 21. ^*-iy = 2. 

jx = 60, 23. r = 2' 35. r^^ 



10 



11 



12. 1^=^' 24. r=^' 

1 y = 4. * y = 7. 



ELEMENTS OF ALGEBRA. 



67 



X = 



2. r 

(y 
(y 

X 

y 

X 

y 

X 

y 

X 



\y = 



6. 



7. 



(y = 

rx = 

\y = 



X = 



y = 



i 
§. 
-1, 

-2. 
-44, 

6, 

8. 

12, 
6. 

am—bn 



Exercise 85. 
9. r = ^' 



15. 



x = h 



11 



12 



13 



<x = . 

\y = l 

{^ = h 

' \y = l 

(x = -l 

iy = h 
(x = ie, 
\y= 7. 

<^ = h 
\y = h 

Exercise 86. 



16. i^ = J' 

(x = i, 



17 



18 



19. 



20. 



21. 



x=l, 
1. 



(y = 



« = iWr, 



J" 

I- 



x = 



y = 



ar = 



y = 



an — bm 




a^-b^ ' 






nq — mr 




I a* + aft + &• 


Iq — mp ' 


6. - 


^- a + 6 


Ir ^ np 
Iq — mp' 




aft 

l^= a + ft- 


be 


7. - 


\x=''^p:. 

qr-p^ 


ac 




., -P9'-*** 


a«4-ft»' 


l^-,v-y 


« + fli 




ax^ — a 


OiA + afti ' 


8. i 


X = Sf 

Oi — a* 


*,-6 




1 — a Ox 
1/ — *. 



axb •\- abi 



Ox — a* 



58 



ANSWERS TO THE 



9. 



10. 



11. 



12. 



13. < 



14. < 



am^n 






h 

+ b 
a 



a — b 

,2 



a' 
{a — c) 



y = - 



a — c 



nr 
X — — , 
c 



X = 



c{a + b) 

2^ 
c (a — b) 

2a 

2n 
m -\- n 

2m 
m ■\- n 



15. 



f m — n 



X = 



m-\- n 



aai (a — ai) 



16. ^ " -r «i 

I _ aai(a + ai) 



17. 
18. 

19. 

20. 

21. 
22. 

23. 



r 

\y 



X = m -{- Uf 
m + n. 



a 
be 



I- 



a + 2b 



x = ~ } 
m 



n 

1 

— > 
n 

L^ m 
r a; = a -|- 5, 
\y = a~b. 
(x = m + nf 
\y = m ~- n. 

a'^jb- a) 
^- a« + 6» ' 



y = 



a^ + 2b^-ab 
b — a 



24. { 



X = zfy 

mn — 1 



y = 



4-1 



mn 



25. 



( X := m — Jlf 

\y = n — m. 



ELEMENTS OF ALGEBRA. 



59 









Exercise 87. 






( 


x = 3, 






x = l, 




1 

x = -, 
a 


"1 

( 


y=2, 

.z=b, 
x = 10, 




4. ^ 
f 


y = 2, 

z = 3. 
x = -5, 


7. < 


1 

1 


"• 


y= 2, 




^• 


y = 5, 




'x = 20, 


1 


.2=3. 




I 


« =5. 


8. < 


. « = 30. 




ar = 2, 




( 


-:r= 7, 




rx = -12, 


'■{ 


y = 3, 

.« = 4. 




M 


.;s= 9. 


9. ^ 


y= 6, 

U = 18. 






a^i 


5 


> 




2 




^-a6 + *c 


4-ac 


~ ?/l + 71 


10. < 




abi 


? 


9 


12. < 


2 


^~a6 + 6c 


4- ac 


^ m-j- p 






abt 


5 


• 




2 




[ a4 + ic 


+ ac 


. ~ w +/>" 










par = 4, 




rx = -li, 




r' = i, 




14. . 


y = 5, 


17. ^ 


;y = 2i, 


11. . 








[v = 3. 
\x = a, 




U = 6^. 




a 




15.. 


y = b, 


18. . 


y = -3i, 




» = 2' 






z = c. 




U = 2^. 


13. V 


/=2- 




16. . 


rx = l, 


19. i 


fx = 7, 

y=9, 

U = 3. 



60 



ANSWERS TO THE 



2. 



3. 



iy = 



3, 

2. 

13, 
5. 



y = 



a + b 
0. 



4. 



r 



z = 



a, 
b. 



6. 



Exercise 88. 

(x = 2, 
13/ = 3. 

^ = ^^^475-2' 
a^bc 

\ X = 5^7^, 

^ ~ abi + aib 

2 a aib 
^ ~~ abi + aib 



11. 






10. 



mn 



12. 



13. 



14. 



^ — ^^ 






2/ 
U 



— 3 5- 



y = 



m 






X = 



15. ^ 2/ = 



z = 



f._ 



m 


+ n- 


Sp 


w 


a 


— n 


n 


b 
+ 3p- 


— m 


m 


c 





20. 



f _ (a + ^) w + 

_ (a + ^>) ^ + 



2, 

3, 
4, 
5. 



^m 
"^X" 



16. 



VJ.\y = 



2 m ' 
mn -\- mp -{■ n p — n^ — 2 p"^ 



z = 



2(n' 
3np — mp - 



p') 



m n 



2 (n^ - P^) 



19. 



y = 

z = 



18. f = 






ELEMENTS OF ALGEBRA. 



61 



X = 



21. ^ V = 



mnp 



mn + np — mp 



2mnp 



z = 



nip + np — mn 

2mnp 
VI n -f itip — np 

ar = ^a + * + c) — a, 



32. 



X 


= 


2 


mnp 




mn -f 


np- 


mp 


y 


— 


2 


mnp 




mp + 


np- 


mn 






2 


mnp 





mn -\- mp — np 

1 



x = - t 
a 



23 



rar=|f (a + ft + c)-a, i 

I z = § (a 4- * + c) - c. 1 

27. f^ = ^^* + 






W* -I- 7i*, 
>8 





30. 



31. 



34. 



X = 105, 
y = 210, 
z =420. 
a; = l, 

y = 2, 

« = 3. 

x= 7, 
y = 10, 
2= 3- 

= 7W,* 



35 (x = m''-n", 



2. J. 
3- h h 

4. ^. 



Exercise 89. 
5- A- 



6. 



a-\-bm b + an 
mn— l' mn— 1 
10, -1. 



7. 16i, 15. 

cm — an 



10. Corn, i; oats, f . 



,, /t bn — dm 

11. Corn, ; oats, . 

oc — ad 'be — ad 

12. 180 lit 2 for 3 cents, 300 at 5 for 8 cents. 



62 ANSWERS TO THE 

100 a (b c d - 12 en - 12 np) 

. 777 r at a eggs for m cents, 

13. ^ d(b m — a n) °° ^ 

100b(127nc + 12pm — acd) 

TT, — c at b eggs for n cents. 

a {b m — an) °° 

15. 245. ^l^L±_?!l!!) 

^^* " 2 a - 81 ■ 20. 853. 

16. 891. , ,, , 

19 ^^^ + M11-^) , 21. 315. 

18. 39. 11 — a — c 

23. A, 120 ; B, 80 ; C, 40 ; altogether, 21^9^. 

. ac(n — m) ^. ac(n — m) 

24. A, ^^ ; B, ^ . 

nc — a mc — a 

25. Arithmetics, 54 ; algebras, 36. 

26. Crowns, 21 ; guineas, 63. 

,., nia^c — a Ci) . n(aic — a Ci) 

27. Crowns, — ^ : guineas, — ^^-^ ^ . 

a^n — am cm — c^n 

28. A, 50; B, 21f. 30. Stream, 2; A, 10. 

. mn ^ mn 

29. A, ; B, 



31. Stream, 



a a -\- n — m 

mhi — rrixh . mhi -\- m^ h 



2hlH '^hK 

32. 39 miles, 8 miles an hour. 33. 5. 

34. Going, 4^; coming, 1\\ stream, 1\. 

hh . ah ^ m (a" — b"-) 

35. Gomg, ^-p-^; coming, ^-^; stream, \^^^ ^ 

m{a-\- by 

2abh ' 36. Sugar, 5; tea, 60. 

r^ 100 nOO (71 - m) -hm'] 

3,J^-^-' TW^) ^ 

100 [am -100 (yi-m)] 
^ea, ^^ (a - 6) 



ELEMENTS OF ALGEBRA. 63 

38. Sum, $1000; rate per cent, 6. 

39. Sum, dollars; rate, -— r— • 

40. First kind, 14; second, 15; third, 25. 

41. Fore-wheel, 4; hind- wheel, 5. 

42. Fore-wheel, ; (rnr-'sn) ^ ^^^^ hind-wheel, 

' (m + 7i) (cs + cr -{• ar) 

b(mr -8n) ^^^^ 

\$ + r) {em + en -\- an) 

„. , . . . bnti — bim aim — ami 

43. First kind, — { \— ; second, — r j- . 

aib — abi Oib — abi 

44. First, 3} ; second, 3. 

^. ^ a(m-\-l) . a (n -\- 1) 

45. First, — ^^ T^; second, — ^ :r * 

mn — 1 mn — 1 

46. 3^8, 80% ; 4's, 125%. 

bd -\- np bd — mp 

4a A, 55; B, 105. 

A ^{n — b) (m -\- n — a) c {a — n) (m -\- n — b) 

*^' ' m~(^r:^) ' ^» 7^7^^^6) 

50. Sum, 500; rate, .04. * 

., o bm — an ^ n^ m 

51. Sum, —7 ; rate, 



b — a bill — an 

52. Pounds, 60; cost, 28. 

53. Larger, 5.678; smaller, 1.234. 

54. Smaller, :j -; larger, - 

1 — no 1 — a 



64 ANSWERS TO THE 



1. n^^ i. 

216 a« 1 
3. 



^ a.^ + 1 

4 ™n + I . 1 . . 



Exercise 90. 




a' 4- /^«'-*'Y 




"•''*' u + w 




^- ^ 




2" ' a' (a'" - b*") 








11. ^"'f'.^ + lZ;^'''-"*-!; 


1 


1 16 








16" a"""* a;""' y^ 



' • m m > 1 27 • 



Exercise 91. 



13. aVa;^ 6^2 6^c2«'. 

14. 2". 15. 22'". 
16. (a;2-y2^-3«j 2«. 



1. ^ ; 1 ; ^a;^ 1. 8. V-{a-2 x) 



9 ^\17(l + a) 



a»' _i_. L_. 1 9. a;2«; 1. 

1 - „- 1 10- (^ + 2/r; ^"-^'^ -6 



3- ^r-;;r; va»; Vet'"; - 



«- 



11. 



_1_ /^\ {m + n)^ 



4. — 4 a/2 ; ^a\ xy. 

6. 3"; (a -5)"-; g- 13. ^a ; 2«'. 

7 ^\'/"^. ^' 14 a- a;« + ^+-^"- 



ELEMENTS OF ALGEBRA. 66 

Exercise 92. 

1. 9; X; 4m-*; -^; ^. 

2. IG; o^^i'^; 516a»''6»»-^c»''-«. 

3. 343 aV"*"*; (27)'»6»'-« Va*"'; a:*- 

5. |,; y»^-^-; 25. ^- S ' 1 5 V^' 

7. _^L^j (25)'"'»a'''r***'"c'^'"^"S J^4. 8. a:"'"; a^'^"- 



Exercise 93. 



1 ,1 ,1 (b-\ 

^- (^-Y <""">' (^»' (''^^'' (;^' i^'i 

Va-'ft-'y-*'' 

_1_ « m 

4. -fo^; ^iO^^ yaHW', yJ a^^ b" ; \ ab^-, ^ x^ i/\ 

6. Sa^bc^', 5a-^b hr^; mlnij ab^; x-^t/-^; ^"*y^; «'**• 

2 • 1 1 1 5 _ 1 

^- 5a-«ca:^ a-J^!^ -^^^ xjri^*^ Sa^J^V^ ^""^' 

rt* 1 

8. «*; jgj «*"•; a»; -; al. 



66 ANSWERS TO THE 



m 



10. -; 7^TT-^', 1- 20. -t; 1; 1. 



12. |;;16.^m;^; 22. 2«%^ 



M , ^^ 23. I; 18X10^^' 

\/nr J/7. ^"^ 3' 



2 »i m t t 



1 6/- 

a; " 

Veto * "? ^ ± 1 

27. a'* — a2»^»4« ^ 523 



a^ 



Z.2 ^ 29. £c2n-2_ 2m- 

17. ao''; fw7^ic. "^ 



t m t 



11 . . 

32. a** — 1 + a~'* ; n^ y^ — n^m^'xiy^ + mTn*xiy^ — m^n^x^y 
+ m^n^x^y^ — m^nrxy^ + ?7i7a3s. 

(as" + &-'*)(ai'* — b-''). 

b^ Ab^x 6a;2 Ax x^ , ^ 12 8 

34. ^ + — T 72 + 745 x^ — 6x + — + -^', 

a* a^ a^ ab^ b^ x x^ 

5n . 5n 



a 2 _ 5 ^-^n _^ ^Q ^-^n _ ^Q ^^n _,_ 5 ^f , 



a 



35. 2! X 3^, 2i X 3^, 21 X 3^, 21; 24. 

36. 2§ X 3^, 2i X 3^, 2i X 3^, 2i, 2i X 3^, 2§ x 3*; 576. 



ELEMENTS OF ALGEBRA. 67 



37. 


ai + i; 26i- 


a-H\. 


43. 


a'-'-^+a; 2^^' 




38. 


2a-\-a-\. 




44. 


77» ^ 




39. 


x-^x-\ 






y' 






1 




45. 


xi;2. 




40. 


X 






,,-^ ,; 2»«% 2. 






X 




46. 




♦1. 


3-2a^-. 






i^m" 




42. 


x-^ -. 1. 




47. 


(5a:^-3yY; m« - 


-w«. 



Exercise 94. 

1. V^i; \^V"; ^64; ^i*""; ^8"^; ^27 aH*c^; '^/W^', 
^; ^^; ^i^^. y/i; y/J; ^/2^; ^F; 

a. Vi; ^i; V800; VH;_v/^¥^; v^^^. 

5. VW^^l^^, v^ii^^l; V/^- 

8. 12 V2; 15\/6; -9v^; | \/58; \a</Va. 

9. |V2; f\^4; ^-^^20; 7^3; iV5;iVl5; i ^32. 

10. i'^'24; VV6; -3a6^4^; 2aJ6«'^; ;r^,\/3a*na;. 

11. a*i-V^c; -V^; — V^^aJ. 



68 ANSWERS TO THE 

12. ^ ^6 ; 9 a" b^"* \/Wl? ; - ^2ax'y'; x y"" ^yx\ 



13. (x + y)Vx — i/; (x — 4:)^a. 



14. 


t>(^ + y)'''' 


7 


^3n^2,n^5^ 


— m 


15. 


72; 3; 2. 






20. 


fa^c; a^oTx. 


16. 
17. 








21. 
22. 


1. 

3 2 

2' «• 


18. 


10 a m X. 
Bab. 






23. 
24. 


2 a. 


19. 


a</4; 2a«Z»'^8a2i» 



Exercise 95. 

1. v^8; ^9; §^1296; 3^64; ^^»; ^8; J ^16. 

2. V3; tv^64; {^«^'; •'^?. 

3. -v/a^?"; y/o^^; y/a^ SJx'yl', ;/a-; y/-^; :.'Y«'- 

4. v^l25, V^m, V^l3; v^i024, v'625, y'MS; -^^Gi, 

^81, ^6. 

5. ^2, ^2, ^^2; ^49, ^625, ^2l6; ^a«, 3^^°. 

6. ^^, ^^; ^a^, ^^^ ^^; ^^, ^^^ 



7. V< ^625^, ^27^; VW^', V27 b^ V^^x^ 

8. a:^< ^^«"^; ^< ^< ^^, ^^^^^^^. 

9. '^a', W, ^Z^; 4^625^«, 2 v'8T^^ 10 a ^729^ 



ELEMENTS OF ALGEBRA. 69 

Exercise 96. 

1. 2Vi4; eVH; 6^4. 3. V^l v^4; v/f. 

2. 10 VS; ^ViO; ^2. 4. ^Ti; 1.6; V^. 

5. V3, -^7, ^4; 8 a/2, 5 a/5, 4V7|. 

6. 4^/7, S'C^iO, 2V^21; 4v'ii, 2^l3i, 3^. 

7. 3V2, 3^5^, t>^; 3\/8, 2^/8, 2^21. 

Exercise 97. 

1. 8\/5; tVlO. 7. c (6 c — a) \/2 a * c. 

2. 2V3; 6^6; | v^2. 8. mnx\/mn^. 

3. -a/3; 17^. 9. 2bVS~a. 

4. 20 J a/3; fA/15. 10. - ^'^^~J\ 

5. ^a/IO; 0. 11. f|A/6-^14. 

6. — Vva*; 2v5. 12. — ^^ s ^a/wx. 

Exercise 98. 

1. 18; 18 a/7; 25 a/5. 6. ^^^486; v^ll8098. 

2. 2; 14 a/G; i >^12. 7. § >^ ; J a!^ ; 14 >J^. 

3. 9v^288; 10 (3- a/3); S. 8. -729; 

,— 6(l-3A/3 + 3ViO). 

4. 2+ A/6 + 2VIO; 120. ^ 

5. vTO; T»B ^2592; ^500. 9. A/ lOO; n-V^; 12v^2. 



70 ANSWERS TO THE 

10. in n ^72li^x\ 19. 5 ; 4 \/32. 

11. m^m^7i^x^^', 6\/^iH\ 20. 4; '^3. 

12. v'72^; 2«m"a;". 21. \/^- Vn-, yVv^lSOOO. 

13. ^V2; -12-1 V^. 22. 2; -v^9^^ 

tt X 

14. 9ViO-12 ^32 + 6^^3125-8^/10; ^^4 + 2 -^6 + v^9. 

15. 4(6VlO-l). 

16. ^^128 - ^^2187 - '^972 + -v^O + v'i - V6 ; 12 ^^27. 

17. - 1 ; 'v/y - 2 ^^2. 23. 49 a; - 9 a. 

18. 2 ; - 7. 24. 1 v^288 ; | ^^80 ; 20 \/b. 



Exercise 99. 

1. 2(a/3-V2); ^5(2^/5+^6); 3^/2-4; 3(VT5-VtO). 

2. 4+V2; -l^yV^ ; _^(3 + 2V6); t^re. 



x—2'^xy-\-y '\/xy{?»x-\-^\/xy — 3y)^ a-\-^d^—x^ 
x — y ' y{x — ^y) ' a; 

4. 2 i«^ - 2 X V^^^^l - 1 ; ^V^(^^- h) { a - Vb-) 

a^-b 

5V5 - VTO + 5^2 + v/8 2 a \/a:'- '"«'-« 



23 ' 3 a; s 

5. 14.14; 7.07; 141.4; 11.18; .1732; .2236. 

6. .707; -.236; -1.266||; 1.216§; .268. 



ELEMENTS OF ALGEBRA. 71 
Exercise 100. 

1. 3V3; |V2; '^3. ^^ axV^-^Vy; — ^^^^^. 

3. 3.3; ^VO; i ^486. "^ "" 

4. 2^3; ^54; ^18. ''• V ^^-^^ ^ ; - ^1944 aV^ 

5. 6^5; i^96; 2^/2. 13- ^^'^^^^5 1 4- V^. 
7. 5 V7 - 4 VC + 2 V5. ^3 5 - V6; ^ '^a^^^ 



.- le/-^ 



3aa; 



8. .2; a^6c; --^aft*. ^ .-^^ ^ , /^ 

* 16. 1 (4+ V15); r)+ V<. 

17. ^x + V^^ + Vy. 



9. !!Lzl!^; 1^96 



Exercise 101. 

1. ^^; V3; 2V3; V^8; 2 A^^i 



2. i>^; i^J^; V«-^; 2. 

3. 4^; ^n»; 2^'/2; V2; 2 15^2. 

4. V^; 3^; H; 4«^3; ^ ^a.' 



5. 8aVft; ^V^i^^^^; v'-^; 32a'bc\/2abc, 



6. 3a; 648^/^a:^ 2a«; ^^ 

7. r,; ^(«-rr; 1; ^^^^ 

4 



aC 



72 



ANSWERS TO THE 



8. JV3^; a'"-V«; va''; -4- 



1 X, 



9. -\^n^—'-'; "''^(aH'^)"^^'-'^^^; m'n' ^/n\ 
n 

1 8/-r- 0^^ iC*' V^^^ ^^ 
10. -,; Va<o; — ^g 



11. 



^.; ^2/(^ + 2/)^ 2a^^3. 



27 



12. x^ + 2;r27/2 + 2/'; t + 4V6; 11 V2 + 9 a/3. 

13. 2 - 3 ^y + 3 v^3 ; 4 + 2 V2 + 6 -^4 + 6 '>^32 ; 



m' 



'\/') 



m^ 



3/— , O 6/- 



iiin 



14. 



_^ ^6 — 4a" + a^"; v<*^ — — 



^58m- 16 



15. 



5 10 10 5 

^12 + a« + a^ 



^^ + ^e + ;;r2+:;F+-4 + l; «*- + 4 a«- 5 + 6 a^- 6^ 



16. ^V2m7i + |--2^m7i2 + ^^8m7iH404- — ^Sm^Ti 



, 32m 3,—, ^ , ... --- , 






Q>A:m^x^''. 



Tn^w 



w 



17. ^ + 1-3^2/. 



19. ^ + «^^2/'- 



2 2/^' 



Y2 



18. 1 — 2" ; 3" — 2\ 
1 



6"/;:;57 



20. ^^a^--2V;i^- + V^ 



ELEMENTS OF ALGEBRA. 



73 



Exercise 102. 

1. ^9; 1.2; 9; 1; i; |. 

2. a»v^2; mv^G; i^768; 4. 

3. ^^5^; ^{/243; 6^3', 4 Vm (m + n). 

4. -V^^?^^; i^xTy; e^^^^v^a. 

5. ^64, ^?^, ^l25; ^6561, '^EV^, i?^15625; 15^2?; 

9^4. 

6. 6^*64, 6^16, 12 ^8, 10-^4; ^a\ ^6", ^^\ 

7. Va, 2V3; 3v^2, 3^2; 2v^9, 3^9. 

8. iV5, a/5; V^5,3a!^5; Vtr VlO, t^^VIO. 

9. §a^9y^2ftv^96■^^^^'9P; ^ ^50, ^ ^50. 

10. l</% \^% ? ^9, ^; 2 V5, ^ V5, 20^/5. 

11. V2, V2; 2a;V2, %bW% « V2. 

12. i ^, 2^6. i ^6, i ^6, i ^6; 2 v^, ^2. 

13. 4^^, 3 \^3, 3^/275-, 3V19, 5^8, 3^9. 

14. 2 ^8, 5 ^2, § V3, i V5; ^H, Vf 

"• ^^^VS^» -^^5^(2i^^,aV(8^^. 

16. 16 V3; 7^7. 20. 8^4; v^3. 

17. 4>^l6l25; 3 15^4; ^624288. 

18. 3^/2; - ^3. 21. H\/15; T^y. 

19. 24v^4; 1. 22. i; 6. 



74 ANSWERS TO THE 



23. |172; 0. 34. 1; i; ^^ ^^. 

24. -/s'^^; ly^. 35. -^^V^; 0. 



25. 156-24V4. ^^ cm 

_ 36. 11; — -. 

26. f(7-2VlO);iVl5. ^"^ 



Vw/^ + «^ ^^ — <^ ^ 1 



27. ^l:::_lj^l^! — ^-. -V^«z/--- _j V2+iV3+^V3o. 

77t y 

28. ^ v^lOSOOOO ; 204.8; ^a'h'', 

29. 47; 27. V^. icB - 2 

30. a^-i + ljar^y-t^n+D.j/^ ^ ' xl + 2 

31. a^^. 39. |V2. 

32. cVc; ^a. «,y^2o X 315 



,.y/^ 



33. c ^c ; h '^Wb, V 2^^ 

38. 1 ; Va (a V^ + 3 a + 3 \/a + 1). 

Exercise 103. 

1. 3V^^; 2\/^; 2V3^xV^; 2aV^; iV^. 

2. 7a"^»V=^; 3'^=!; a^^/=l; 2'^-^. 

3. -1; -V=^; -a/=3; +1. 

4. +V^; +1; -V=^; -V^=^. 

5. _4Vin[. 8. 2&(2a2j-l) V-^. 

6. 4 V^ - i V^3 - 3 V^^ ; 0. 

7. a2(8a + a2_^y'ZrX. 9. -3V2; -36^/3. 



ELEMENTS OP ALGEBRA. 76 

10. -48; 2,^/^-i, 16. 6»-a» 

11. ~GV<i — ^9. 17. V3j AV3. 

12. 39-2V^; 21. 18. i; 2V2-\/3. 

13. 2; — 4 — 5\/^^- 19. 4; - Va — V^. 

14. 30 + 12 V2. 20. 1 - V^; 1 - V^. 

15. (a - h) V^^; a + «. 21. i A^2625; V3. 

22. 1 + V^^; \ V^^ - i a/3. 

6 + 2V^ + 3\/^~a/B 24 + 10V3-15V6-8V2 

^- 3^6 ^ 43 5 

lV-1. 

^ / — T- a* — 2 a V— « — a; 2 \/a^ — a — x 

24. 1 — V — 1 1 7. : } • 



Exercise 104. 

1. V5-A/2; V3 + 'v/2; 4\/2-3; iVB + 1. 

2. VI0-2V2; V6 + V5; V7 - \/6. 

3. V14-1; 2Vll-V'3; V^-l; V6 + 2. 

4. 2-iV3; V^2 [a/3-1]; a/5[V2 + 1]. 

6. m Vw — V^ w; a/'J' -- a/2. 

7. m — n — 2Vmn; /v/^('y/2 — 1). 

8. 2-V3; i(V5+l); a/5 + 1. 

9. a/2 + 1; V5-V3; V^(V5+V^). 



76 ANSWERS TO THE 



Exercise 105. 

1. 11; 10; -f. ^^ (x = 7^-^VTS, 

2. 4; 15. * (2/ = |A/i5. 



c — 



1 9n^ 
12. —; 

34 m 



^- ^^' "^'^^ f 2V«(V«+ v^) 



iC = 



a — b ^^ I "^ I) — a 

; 54. 16. ^ 



I ^ ^ 2 (Va6-l) 



5. 2; -,V '^ ^-^ 



6. 192; 3^. ^^ (^ = -li, 

7. 4. 



8. 1. f _ Q^ — ^ 

18. ^^-l-n' 

^^' 4m '^' ^i- r m + y.-^ 



- } 



13. 15; 6. ^^ I Vm 

14 ^(^ + ^) . 



h 



Vn 



Exercise 106. 

1. 0.7781; 1.8060; 1.1461; 0.9030; 1.0791; 1.1761; 1.9242, 

2. 2.5353; 1.2040; 2.3343; 1.4313; 1.6532; 1.5562. 

3. 1.9542; 2.3222; 3.5562; 3.0491; 3.2252. 



ELEMENTS OF ALGEBRA. 77 

Exercise 107. 

1. 4; 1; 2; -2; -1; -1; -3; -7; 0. 

2. 0.8281^ 2.8281; 1.8281; T.8281; 3.8281; 6.8281. 

3. "Six;" "one;" "four." 

4. "Fifth;" "first." 

5. T.2552; 1.3522; 0.0212; 0.5741; 1.0212; 0.7993; 

2.0970; 2.6232. 

6. 3.7481; 1.1070; 1.1582; 0.0970; 1.0970; 2.6990; 

5.4983. 

7. T.4804; 0.7323; 5.7781; 3.3222; 0.5441; 4.5441; 

0.6511. 

Exercise 108. 

1. 1.2040; 2.0970; 2.5350; 1.8060; 3.3397; 1.8060; 

1.9084; 1.8572; 6.3588; 1.6110. 

2. 2.5353; 2.5562; 4.1070; T5.2620 ; 3.5810; 5.2569. 

Exercise 109. 

1. 0.3980; 1.7781; 0.1461; T.7781 ; 0..5441 ; 5.6320; 

T.3980; 2.4950; T.5441. 

2. 0.6690; 4.1373; 0.0970; 3.7781; 1.5899; 4.1040; 

0.6990 ; 0.4559. 

Exercise 110. 

1. 0.1690; 0.1590; 0.0602; T.9466 ; 0.7952; 0.6476; 

T.9964 ; T.8950. 

2. 0.8063 ; 0.3523 ; 1.2519 ; 0.0691 ; 0..3093 ; 0.5456. 

3. T.6360; 1.6507; 0.2605; 0.6851; T.9962. 



78 ANSWERS TO THE 



Exercise 111. 



1. 1.8451; 2.0086; 2.3032; 2.9996; T.8525; 0.5563; 

S.8971 ; 0.5065. 

2. 3.4914; 2.9926; 5.4771; 1.0034; 4.4692; 3.4794. . 

3. 4.5142; 1.2638; T.5876; T.6235; 0.7672. 

Exercise 112. 

1. 2940 ; .0289 ; 63900 ; 3.151 ; 1.57. 

2. .00455 ; 30.94 ; 33138 ; .03333 ; 4.566. 

3. 7.586 ; 50.56 ; 633420 ; .001301 ; 539375. 

Exercise 113. 

1. 29.77 ; 4.75 ; 2.814 ; 32464. 

2. 2.664 ; .368 ; .0769. 12. 9 ; 20 ; 121 ; 30 ; 56. 

3. 373.6; .4847; .6186. 13. 2; 30; G; 14; 1. 

4. 1.683; 2756; .482. 14. T.75729. 

5. 9745; 3.264; 3.637; .4276; .2163. 

6. 1.53+ ; 9.58+; 6.44-; 5.59+. 

7. 2.56+; 1.2+; 2. 15. 2; 3; 5; 7; 10. 

8. 4.56+ ; 3.46+ ; 3. 16. - 2 ; - 3 ; - 5 ; - 6 ; ^. 
log n log n — b log m log n 



9. -3.3+ 



log m ' a log m ' a log m -\- b log c 

'09+, \ X = 1.177-f 
709+ ; '1?/=: .677+ 



^^ |a::::.2.709+, 1 ^ - 1.177+, 17. g ; 3 ; 4 ; 6 ; - 1. 
\y = 1. 

18. 4; -3; 3; 6. 
_ 4 log m 

"^-"To^' (x = 3, 19. 2;|;i;-4. 

\ _ ^Qg^ . h = ^- 20. -y; -t;-2; 1; 

^~ log 7^' -1. 



ELEMENTS OF ALGEBRA. 79 

21. 4; 16; 8; 64; 512; 32; 256; 1024; (2048)\ 

22. .00001; 5^3; 243. 

23. 3; i; .3; V; V> and t?^; 4, and 4. 

Exercise 114. 



1. ± 3; ± 2\/A- 8. ± Vm + n ; ± c. 

a. ±2J; ±Vi|. 9. ±\/^i ±V^f^' 

3. ±8; ±5.5. 1^^ 

10. ± V t; ±4. 

5. ±.3; ±tV2. ''»-gl 

6. ±2; ±^. "• ±"i±v!! *• *^- 

7. ±|V30; ±}V77. 12. ±\/*(6»c-2«;; ± J. 

" c 

Exercise 115. 

1. 12,-2; 8, -10; 16, 2. 7. ff, - y ; 0, 4. 

2. 17,-4; 4,-13. 8. 2,-4,-5; 1,-1,-^. 

3. -3, -i; i, -3. 9. 2, -V^; |a, ^a. 

4. 107,-106; §,-i. 10 '',-^; 4,-1. 

5. ^,-2; 2a, -8. 

, o 11- 1.-^5 ±5, ±3\/2. 

6. 0,-3; T^, J. 'a -6' 

Exercise 116. 

1. 8, 15; 7, 6; \,-\. 5. -?, -i; V-, -^3^- 

2. 23,-1; 6, -J/; 1,-4. 6. V, -§; A.-?- 

3. 2,-V; 3,1; 3,-1. 7. 1,-3; i,-J; ^^-J. 

4. 11, V ; h -4; f -A. 8. V, -^\ ^i -f 



80 ANSWERS TO THE 

15. i(5±V22:6); 8,-^. 

16. 11, 2; 7, 2. 

17. 5, -4f. 

18. 9,-H; 3,-3§§. 

19. 13, ±VM; 12,-2. 

Exercise 117. 

5. 71)— p\ ± ^2n — ^/n. 

/ — ;- ha 1 

2. ±yah\ -, — T- 6. a ± -; ^, — a. 

a a 

^ . "^h b 2b ^ . a 

3. 1, - — -; -, Q, Za, —a, a, —-' 

a — b a a 2 

a^ __2y 8a a 5 =b V25 — 4:m^ 

a(a + ^) a (a — b) 



9. 


3,-t; -i,-6§. 


10. 


1,-14; ±9. 


11. 


13, §; 9,-ff 


12. 


3, -8.7; 6, e. 


13. 


2, ^Y-; 8± V601. 


14. 


12,-2; 3,-i. 


1. 


, « b 



7. (a^± ^»)2; 



a + b 



9. (a + ^)^ - {a - by; 0, ^ 4- - - c - 1. 

Til 

10 ^ + ^ <^-^ . a ± Va^ + 6'^ 
a — b' a + b' Sa^b^ 

a f ^b^ - 4 c'' \ l — 8a± 2aVa 
^^' 2 V "^ 2«^ + ^ ; ' (a - ly 

13. ± 1, ± ^ V^3; ± V2, ±^V6. 



ELEMENTS OF ALGEBRA. 81 

Exercise 118. 

1. 9H,-11; 2,4. 4. 2, i; 3,-2. 

2. 7,-7|; -i,-?^ 5. i,-i; 3, f. 

3. 4, -3S; 2,-8. 6. 4a, -^ a; 2,-22. 

8. |(-1±V^); ^,-a,^(-5±Vr4); ±V;^-1. 
a -\- d ac + bm^ 

»• 1' — d-' "' ^ — 

10. — , ^; m — 2 a, ^ m -\- a. 

n m, 

11. a ± (6 — c). 16. 2 rt + 6 *, a — 8 ft. 



2 * w — 2n' m ■\- 2n' 

~^^\ -1 ± ' 
a + ft ± a/2 a^ + 2 ft* 



13. ± V2 a - a^ -1 ± A/T^=^; - (2 ± Va^ + 4) 

a 



14. 



4,/ _ 5ft a -2ft 2 /J7 

15. ^ . , » . ' 18. ±^V3. 

6aft 3aft 3a 



Exercise 119. 
2. 8, 9. 3. 15, 12. 4. 6, 21. 5. 8, 6. 



7. 



,»iO±\/^)- ^1^- 



82 ANSWERS TO THE 

9. Greater, _-«y»L^; less, _^V" _ . g. g. 

10. 18. 13. 87. 14. 53, 35. 

11. Eate, 30 miles an hour; 7. 12. Kate, 6 miles an hour. 



2m — n ±. ^/ii^ + 4 m? 

16. La]'ger pipe, ^^ minutes ; smaller, 

2m^r n± V^ + 4 m^ • ^ t • oo • 
minutes. Larger pipe, 88 mm- 

utes; smaller, 154 minutes. 

17. B, 6; A, 10. 18. 9 miles an hour. 



19. miles an hour. 4 miles an hour. 



14: am „ 
-a± V^^ + ^ 

20. Length, ^— ^^^^' 

weight, /, lbs. Length, 8 feet; 

^ A arn, „ .,,.-,, 

— a±\/ \-a^ weight, 4^ lbs. 

21. <^VnTmVc Vn(VnTVc) ^^^^^^^. 

^Jn^ \/c a-m 

Vc(Vn^:Vc) ^^^^^^^ -^g ^^yg . j^ ^1 25 ; B, $1 , 
a — m 

22. 10 (— 5 ± Vfn + 25) dollars. 



23. 10(5 ± V25 — m) dollars. 

Exercise 120. 

1. ±2, -fcVlO; 1,-2; 2, -1, -1± ^=3, ^(1±a/^). 

2. ~3; ± a, ±5. 4. 16, ifffi-; 8, W- 

3. ±4, ± i; 9, (-41)1 5. ± 8, ±(-11)^; 9, ^/i681. 



ELEMENTS OF ALGEBKA. t 

6. 3», (-28)1; (J)«, 1. 9 i JL .>_,., , 

tt- 4 a- * ^ ^ 
'■ '«.li 27. , 1 

8. jm;«)u. -1,(196)". Venn (j^ 

10. ±^^/2^, ± \^^; ± vS, ± v'-S. 

Query. Wliy, iu 10, the ± sign ? 



Exercise 121. 

1. 2, - 3, -1 ^ 3V5 . 9. i(3 ± V5), i(9 ± ^f^^\ 

2 5± V37 5± V7 

2. -1, -1±2V15. "' 3 ' 3 ' 

3. ± 4 V6; 18^, 5. ^^' -' ^' _ 

- 3 ± V33 

4. 3, -i, 4 C5 ± V1329). 13. 1^ , -1,-2. 

5. 5,-6, K-l± V377). 14- 1, 2, -5, 8. 

®- ' ^' 3 16. 5, - 1, 2 ± V5. 

7. 1,-3±2V2. -3±a/29 -3± \/I^ 

— ^'^' 4 ' 4 

8. 4, 1,^^^. 18. 2^e^, ^rs. 

10. 3, —1. 
19. ^-^^, -1± V2; 3,-5, , -2 (-13 ± V313). 

Exercise 122. 

1. 10,-2; 7, 14. 4. 14, 2.48; V- 

2. 9, -12. 5. 4, -21; 5, \. 

3. i, V; 8. -I- «• ^' 4' -^5 12, 4. 



84 ANSWERS TO THE 

7 _|_ 9 V2; 3, ^. Query. Is ^ a true value for x in the 
second equation ? Why ? 

8. 0, ± V3. 12 i^ ^- 4 B4 

2m^/n ^ ^ 

9. 2^T. §; ± ^^ _^ ^ • 13. ± (^; 9, V- 

10. 0,4/7; ±Vl±h^2. 14. 1, 4; 9, 4, ~ ^ ^ " ^ ' 

11. ±*-'(^')'; 25, -V- 16, 0, 2,-3; ± ^^ V3. 
_ 7±Vl3 -lq=V^^ ,, , o-^o./^ 



.u. 


2" ' 2 ) -^'^j -^j ^ -r- 


^ V t , 




Exercise 123. 




1. 


-8,-9; W, 49. 




2. 


. J-;;, 


mn 


m — n^ 


3. 


0, ^; a, a^ 15, -3. 





2 b, y'-a'. 



5. a;2_4a.^21; Gcc^ + Sx^G; a;2-2a; = 15; x' = -Z\ 

6. a;2 + 5a: = 0; x''-Ux = —29', x^ — 2 x = 1. 

7. ic'-^ — (1 + m^) X = m {1 — rn^) ; mnx^ — (m^ — ?i^) a; 

= m w ; (a — Z*) ic^ — 4 tt 6 ic = (ct — ^) (a + hf-. 

8. ic^ + 2 m a; = 8 /^ — m- ; 4 ic'^ — 4 a ic = Z» — a""^; 

Vox^ — '6\/'(ix^h — a. 

Exercise 124. 

1. Imaginary ; real and equal ; imaginary. 

2. Eeal and rational, and different; imaginary. 

3. Real and rational, and different; real and surds, and 

different. 



KLEMKNTS OF ALGEBRA 85 

4. Real and equal ; imaginary. 

5. Imaginary ; real and different, and surds. 

6. Real and equal; real and rational, and different; real 

and rational, and different. 

7. Real and rational, and different, real and rational, and 

different ; real and surds, and different. 

8. Real and rational, and different; real and surds, and 

different. 

9. Im;i<^nnary; real and rational, and different. 

Exercise 125. 

1. ±8, ±(-ll)i; h -i; h 4". 

2. 1^, T^; -1, -ih' 3. I 450; 4, f 

^"/ l ± \/l3 ^/ l ± V- 7 J ± Ks ± 2 \/4 g + 1 
V 2 ' ^ 2 ' ' 2 

5. ± 3, —- ; 5, -4, 

6. ± h h h 7. 4, 1, 3, 2. 

9. ± 2 \/2, ± \ a/=^, ± i V^185 ± 29 Vil. 

„ n 3±4V3 , , "• 4m», to'; -8,-4. 

X2. 0, -^— , 1, 4. ^^ ^_ J 

13. 4, 1; J, y. 16. 0, 1, 3. 



86 ANSWERS TO THE 

17. -a±-— ^=; 4, 3^3. 

18. (i.) a/^^ — 4 ^ a surd ; (ii.) ^^ _ 4 ^ positive; 
(iii.) ^2 _ 4^ negative; (iv.) yl^-4j5 = 0; 

(v.) J5 positive; (vi.) B negative; 

(yii.) B = --. 



j^ .'±V4j^+a- -|,^g 



Exercise 126. 



12/ = 5, 4. ■ j2/ = 9,- 



2. 



r.x = S, -1, J a; = 7, -5, 

l2/=i,-2. ■1j, = 5,-7. 

.^ = 6,-4, 11 \x = %-5, 

'• ly = 3,-7. ly = 5,-9. 

p-= 6,^^,-41, p = -7.4,1, 

*-J3, = 3i,-4. 1^ = 7.8,5. 

. rx = 18, 12J, <x = ll, -8, 

=-iy = 3,-2i. "•1y = 8,-ll 

^•iy = 6,3. "•1^ = 1. 

p = 4,3, ^^ |. = 5,4, 

|y = 3, 4. l.y = 4, 5. 

(x = 8,17§, ig fa; = ll,-7, 

°- l2, = 6,-13i. l3/ = 7.-ll. 
Query. In 14, has x and y two values? Why.' 



1. 



ELEMENTS OF ALGEBRA. 87 

Exercise 127. 

(x = ±5,±7y |a: = 3, -3, 2,-2, 

U=±7, ±5. • -^2/ = 2, -2,3,-3. 



(X= i, 



7, 4, f X = 5, — 5, 1, — , 1, 



7. * j y = 1, - 1, 5, — o, 

p = 3,2, (x=±5, ±3, 

^- U = 2,3. ''• |y=±3,±5. 

J a: = 5, -5, 3,-7, j x = 10, 4. 

*• 1^ = 3,-7,5,-5. • 1//-4,10. 

ra; = 6, 4, ^ ,3.= ± 2, ±3, 

' (y = 4,6. • |y=± 3, ±2. 

p = 6,-6,5,-5, (0^ = 5,3, 

^- |y = 5,-5,6,-6. ^^- (y = 3,5. 

Exercise 128. 

r x = ± 2, ± 1, 
• ly=±l, ±2. 

rx=±2, 
<x = 



1. 


{" 


±3, 
±2. 




2. 


!;: 


±6, 
± 2. 




3. 


i;: 


±3. 




4. 


\l' 


:±2, 

: ±3. 




5. 




■■±h 
■■±h 


±i. 

±3- 



7. 



± 7, ± >v/3,_ 
±2, TSV3. 



( ar = ± 3, ± 36, 
• -j y =: ± 5, :f ¥• 



r X = ± 8, 
1 y = =F 5, 



^^^x=±8,±3. 



88 



ANSWERS TO THE 



1. r 



2. 



3. 






5. r 

(y 

6. f 

7. f 



3,5, 

o, 3. 

6,-3, 
3, -6. 

8,-7, 
7,-8. 

10, 12, 
12, 10. 



10. 



11. 



12. 



13. 



Exercise 129. 
14.]^ 

\y 

15. 



(2/ = 



y 






16. < 



5, -3,l±iV-88, 
-3, 5, 1 T iV--88. 

3,2, 
2,3. 

, o -13 ±V377 

4, ^, ■ -^^^ , 



{y = 



± 7, ± 5, 








± 5, ± 7. 






a: 


2, 




17. 




3. 






y 


20, 








15. 
0, 5, 1, 




18. . 


(0! 


0,1,-1. 




19. . 


P 


±10, 




liS/ 


±4. 




20. . 




^Vf 




21. < 


a; 


T 1, T 2 a/=3, 




2/ 


± i, ± V- 


-1. 






0,4, 
0,5. 




22. . 


[y 


0,-1, 
0,-2f. 




23. . 


y 



y = 2, 4, 



= 4,3, 



= 3,4, 



6 
-13:Fi 


/377 


6 


7± V- 


295 


2 


7:f V- 


295 



2, 


-2J, 


4, 


-4|. 


6, 


-2, 


2, 


-6. 


2, 


h 


2, 


16. 



5 IF V15 
2 

5± Vi5 



±iv/l8±2V-li 
i(l±V-15). 



1,2, 
3,1. 



ELEMENTS OF ALGEBRA. 89 



26 



27 



f 1 -\- ab± V(« + 1) (fl^ - 1 ) (b + 1) (6 - 1) 

^*' j «& - 1 ± V(a + 1) (a - 1) (b -\-l)(b- 1) 

1^^" a-b 

rx=±9, ±3, (:r = 2, i, 

-*^- -iy=:±3, ±9. • -1^ = 3,-24. 

( X = ± 5, ± 2, ± 2 v^, ± 6 V^, 

1 2/ = ± 2, ± 5, ± 5 V- 1, ± 2 V-H. 

(x = ±l, ±3, jx = 3, 2, -3± v^, 

(y= ±3,±1. • 1y = 2,3,-3T V3. 

„ ( « = ± 2, ± 3, ^ a: = 7, 1, 4 ± 2 V7, 

''"• 1y=±3,±2. • ■iy:=:l,7,4:f 2V7. 

30 -^ 

■[y = °a^V3),j(i.-L). 

rx = 2,4,3q:v'21, ( a: = 5, 4, 

^^- |y = 4,2, SiVai. ■<y = 4,5. 

32. r=o'"!' •«>r==*^!'*l' 

(y = 2,-4. <y = ±4, ±9. 

^^Tft' ■1y = l,9. 



33. 



35. 



„=±?1±*!. „ 5a; = 62'S,l, 

^ * a-4 "• (y = 1,625. 

x= ±4, itV^^eB, -va6±V34lo), 



90 ANSWERS TO THE 



^x=± (a-h), ^x = h-h 



^ =1,5. 50. 



2/= ±^' 



ly = ^, 

46. .^2/ -^,5,-,, 

47. j^ = ^^' 52. 9 and 7. 

Query. In 47, how many values has x and y ? Why ? 



63. ± }^V^((^ + ^) and ± ^ V2 (tt - 6). 11 and 7. 

54. 36. 

a + ^-3a^ T 2 V2 (a"" + bj 

55. __zi_I Z_ r — V — 2_y ^^^ 

Z 

a + ^- 3 ^2 + 2 V2 (a^ + Z>) Q ^ 1 1 1 Q 
— - — ^^ . 3 and 1, or 1 and 3- 

Z 

56. — a and — 2 a, or 2 a and a. 

57. 5 and 2, or — 2 and — 5, or ^ (3 ± V— 67) and 

i (- 3 + V=^67). 

58. 6.4, or 4.6. 59. 3, 15, and 20. 



2 c" • , — a ± V4 c^ + a^ 

61. =:^= and . 8 and 4, or 

. — a ± \/4: (§ + a^ 2 

— 4 and -8. 

62. f , or -^f. 60. 4 and 13. 

63. Time, 7 years, or 6 years ; rate, .06, or .07. 

64. Principal, $10400; rate, .05. 



ELEMENTS OF ALGEBRA. 91 

Exercise 130. 

^ (x =11,8,5,2; x = 2,7,12,17, ....; a; = 8, 16, 24, 32, . .. 
^' 1^ = 1,3,5,7; ^ = 7,21,35,49,....; i/ =5,8, 11, 14,.... 

^ a; = 42, 31, 20, 9; x = 215, 202, 189, 176, . . . . , 7. 
^ -^y = 4, 9, 14, 19; y = 5, 14, 23, 32, ...., 149. 

X = 8, 25, 42, 59, ,.. .; x = 7, 16, 25, 34, ....; 

x = 4, 17, 30, 43, .... 
y = 7, 22, 37, 52, ....; y = 10, 23, 36, 49, ....; 

y = 2, 11,20,29,.... 

^x = 6;x = 9;x = 0. ^x = 10, 

|y = 3; 2/ = 3; y = 15. ' \y = 5. 

IX = 3; x = 59; x = 476. ^ x = 4, 

ly = '2;y= 1;3,= 19. " ^ = 24. 



6. 


r X = 11 ; X = 37. 
jy=18;2/=13. 

Exercise 131 


11. 




2. 


59. 


6. 


\ and I 


3. 


13 and 1, or 4 and 8. 


7. 


131. 


4. 


72 and 70. 


9. 


Nine ways. 


5. 


( 5 foot rod ; 19, 12, and 5. 


11. 


No. 



foot rod ; 4, 9, and 14. 

10. Two ways in each. 39 and 6. 

12. A gives B thirteen 50-cent pieces, and B gives A forty- 
five 3^ cent pieces. 

r Horses, 1,3, 5, 7,.... 
(Sheep, 8,23,38,53,.... 



92 • ANSWERS TO THE 

rl6, 15, 14, 13, ....,at^25. 

14. J 2, 5, 8,11,...., at $15. 

157, 55, 53, 51, ...., at $10. Sixteen ways. 

r 6, 13, 20, 27, 34, at 30 cents. 

15. \ 45, 35, 25, 15, 5, at 45 cents. 

1 24, 27, 30, 33, 36, at 80 cents. Five ways. 

(-74, 73, 72, ...., at 20 cents. . 

16. ^4, 8, 12, .... , at 35 cents. 

I 72, 69, 66, ...., at 40 cents. Twenty-four ways. 

r23, 16, 9, 2, at $1.50. 

17. \ 13, 16, 19, 22, at $1.90. 

l 4, 8, 12, 16, at $1.20. Four ways. 

18. Four ways. 21. 269. 

19. 4 at $19, 8 at $7, and 8 at $6. 

22. The first has 63 or 23, the second 37 or 77. 

21, 23, 25, 27, poles 7 feet long. 

23. ^ 18, 13, 8, 3, poles 10 feet long. 
1, 4, 7, 10, poles 12 feet long. Four ways. 







Exercise 132. 


1. 


a;>i|; a^ < 4^- 




12 1"^^^' 


2. 
3. 


x>-^\ X <b- 
x>l. 


-1. 


13. f>^^ 


9. 


x = 6. 




ly < a-b. 


10. 


X = 4:. 




15. 76 or 77. 


11. 


x = 5. 




16. 60 cents. 



ELEMENTS OF ALGEBRA. 93 



Exercise 133. 
8.i^2. a -\- b 2ab m n ^ 1 .1 

5. '^±y>t^. 3(l + a'^ + a^)>(l + a + a«)^ 
X — y X* — y^ ^ ' 

6. If«>/;, a» + 26«>3(^6^ :^ > :^; V2 4-V7 

_ v6 v3 

> V3 + V5. 

Exercise 134. 

5. 4x^-3x4-1 > 6. 18 j2^<^' 

6. 4-2a-'+;i2>3^^^ ' < x > 0. 

7. 3 + A>a^ + 3:c«. 19- j^Jj^^^' 

8. 3(a:-y)^>3(ci4-<^r; 1^ (x^ + y) > a;« - y» 

10. lln'»>a2 + 8 6. ' \x >V- 

11. w^ < 54; „ c < 7i«-^2^ ^ a; < _ ^, 






12. -8>-27; 15>8;5>3. 

13. -3>-4; 4m2 + 1>7i. 24. 2. 

14. w > w ; 2 > — 4. 25. 6, or 7. 

15. 2aj>m-7i; -««>/. 26. 16, or 17. 

16. x< 2.9; X <3i. 27. 19. 

17. jc<|; ar<i. 37. 13. 

22. x> 5, ar<^; a;>-i, x<-§. 

23. a; > §, X < i ; x > 7, x > - 3. 
28. 2, 1, 0, -1, -2, -3, -4; -8. 



Vm ^ Vn 



miles. 126 miles. 



94 ANSWERS TO THE 







Exercise 135. 






1. 


-49. 


5. SO a — 79 b. 


9. 


12. 


2. 


161, 245. 


6. -1,0. 


10. 


19. 


3. 


16,9. 


7. i. 


11. 


-95. 


4. 


98, 243.6. 


8. 103. 
Exercise 136. 


12. 


2, 21, and 2^. 


1. 


779. 


4. If (9-.). 


7. 


76 a 4- 57 h. 


2. 


- 5569|. 


3.--^ 


8. 


8^. 


3. 


a^{4:-a). 


6. a7i(?z+l) 


9. 


13. 



10. (1) 71 = 16, d = — l. (2) 7i = 7, d = 2 a. 

11. 5^, 6^, 7t, etc. 13. 7500. 14. -5,-1,3,7,11. 

Exercise 137. 

1- -^H'-6A, ••••, -^T-V 5. x''+l-x,x^+2-2x,....,x. 

7n -4- 7i 

2. 6.4, 5.6, ....,-5.6. 6. , , . 

3. 4 ?w- — 5 ?z, 3 m — 4 w, — , 6 7i — 5 m. 

4. _2l,-3§, -4f, -5|. 7. -101,-7^. 



1. 48; 384. 



8. ^^W, Hi W, ••.. 



Exercise 138. 




3- — ^t; — 2T^T- 




4. 128; 1. 


6. x^»'- 


,-288. 32^ 
^' 243' 


« (§)'• 



ELEMENTS OF ALGEBUA. 95 





Exercise 139. 






1. 2|f 


4. V(3+V3). 


7. .'5. 


520. 


2. mh' 


5. H3'"-l). 


8. 2. 




3. mh 


6. §(1-4'"). 


9. 6. 


5, 30, 180. 



Exercise 140. 

1. 42; aH^', J V6; H- 5. 40, 16, 6§. 

2. ik; 6x2 — ox — 6. g _7^ ^, _|, 1^ _^, ^^. 

3. 20, 80. 8. Arithmetical, f . 

4. i, i. 9. 2 and 8. 
7. 6, 18, 54, 162, 486, 1458, 4374. 

Exercise 141. 

1. -4. 11. 1,9. 

2. ^. 12. 20,5. 

3. — ij't, — 5V' — ^> — ' iV« ^^f) — 

4. 4, 2,$, I, f .... 13. 10,12,15. 

5. If 14. i,i,i, 01-^,1, -J. 

6 ''4,6. ^_,_^ 2a6 

15. —^, Vab, 



8. f , 5' A, h W, A. !«• 2550 yards. 

9. 6, 12. 17. 3, 6, 12, 24. 
10. 6, 2. 



96 ANSWERS TO THE ELEMENTS OF ALGEBRA. 





Exercise 


142. 




1. 


54 6 : a. 




10. 


-13. 


2. 


9:7. 




11. 


2, or 3. 


3. 


10 : 9. 




12. 


7: 2. 


4. 


(l-y)(l + x):l+x\ 




13. 


an — hm 


5. 






14. 


m — n 
h in — an 


6. 


5a:4; Ay : 6x\ Sy : 


X. 


m 


7. 


28 ic : By, n — 1 : 2a. 




15. 


4:7. 


8. 


7 : 8, 31 : 36, 41 : 48, 5 


:6. 


16. 


"^ah. 



9. a" — h^-.a"- -\-h'- >a — h\a^-h. 

Exercise 143. 

3. 4; 2; t33-V210; 12; ^2^2^ 29. h '. a. 

4. ^2 _ ^2 . 300 ^8 j^^ g^ ^^ 

5- M; 3r\; .8; 4^; 1^. 3-,^ 4 . 1^ 



a?" 



6. 3"; (a + ^)2; ^f--,-.- 32. 17:7. 

7. 15; i; 2^3^; 6^-JZ». ^^- '"^ * ^• 

8. 5/;2; 1. 34. 5: 4. 

21. a = c' c?3, ^ = c^ (^i 35. V&xy. 

23. a.= t^^a^^;a:-3,or-l. 37. ^ = ^' ■ 

^^ ' b n^ 



24. 



( a; z!r i 4, i- 6, ^^a + bV „ . 

^ ^ / 39. £c = ^; — ^7 ); ir = 3,or— 1. 

(y = ±6, ±4. W^y 



27. A invested $3000 ; B, $3500. 

38. Bate of slow train ; rate of fast train : : 1 : 2. 



or THE ^ ^ 

f UMIVERSITY I 

\ K^ ^^ :/ 



A 



*;: - 



(■ 



THIS BOOK IS DUE ON THE LAST DATE 
STAMPED BELOW 



AN INITIAL FINE OF 25 CENTS 

WILL BE ASSESSED FOR FAILURE TO RETURN 
THIS BOOK ON THE DATE DUE. THE PENALTY 
WILL INCREASE TO 50 CENTS ON THE FOURTH 
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OVERDUE. 




SEf 20 



\^ 



SEP 1* t»^ n 



= 17Nov'57HIVI 

peco ^ 

«0M 8*^ 






/ 



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a a